##### Class 10^{th} Mathematics Tamilnadu Board Solution

**Exercise 4.1**- The rates for the entrance tickets at a water theme park are listed below: Write…
- There are 6 Higher Secondary Schools, 8 High Schools and 13 Primary Schools in a…
- Find the order of the following matrices. (i) (ccc 1&-1&5 -2&3&4) (ii) (7 8 9)…
- A matrix has 8 elements. What are the possible orders it can have?…
- Matrix consists of 30 elements. What are the possible orders it can have?…
- Construct a 2x2 matrix A=[aij] whose elements are given by (i) aij = ij (ii) aij…
- Construct a 3 x 2 matrix A=[aij] whose elements are given by (i) a_ij = i/j (ii)…
- If A = (lrrr 1&-1&3&2 5&-4&7&4 6&0&9&8) , (i) find the order of the matrix (ii)…
- If a = (ll 2&3 4&1 5&0) , then find the transpose of A.
- If a = (ccc 1&2&3 2&4&5 3&-5&6) , then verify that (aT)T= A.

**Exercise 4.2**- Find the values of x, y and z from the matrix equation (cc 5x+2 0&4z+6) = (rr…
- Solve for x and y if (2x+y x-3y) = (r 5 13)
- If a = (rr 2&3 -9&5) - (rr 1&5 7&-1) , then find the additive inverse of A.…
- Let a = (ll 3&2 5&1) b = (cc 8&-1 4&3) . Find the matrix C if C = 2A + B.…
- If a = (c 4-2 5-9) b = (rr 8&2 -1&-3) find 6A - 3B.
- Find a and b if a (2 3) + b (r -1 1) = (c 10 5) .
- Find X and Y if 2X + 3Y = (ll 2&3 4&0) and 3X + 2Y = (rr 2&-2 -1&5) .…
- Solve for x and y if (x^2 y^2) + 3 (c 2x -y) = (r -9 4)
- if a = (ll 3&2 5&1) , b = (lr 1-2 2&3) o = (ll 0&0 0&0) then Verify: (i) A + B =…
- If then verify that A + (B + C) = (A + B) + C.
- An electronic company records each type of entertainment device sold at three…
- The fees structure for one - day admission to a swimming pool is as follows:…

**Exercise 4.3**- Determine whether the product of the matrices is defined in each case. If so,…
- Find the product of the matrices, if exists, (i) (2-1) (5 4) (ii) (cc 3&-2 5&1)…
- A fruit vendor sells fruits from his shop. Selling prices of Apple, Mango and…
- Find the values of x and y if (ll 1&2 3&3) (ll x&0 0) = (ll x&0 9&0) .…
- If a = (ll 5&3 7&5) , x = (x y) c = (c -5 -11) and if AX = C, then find the…
- If a = (ll 1&-1 2&3) then show that A^2 = 4A + 5I2 = O.
- If a = (ll 3&2 4&0) b = (ll 3&0 3&2) then find AB and BA. Are they equal?…
- If a = (rrr -1&2&1 1&2&3) , b = (0 1 2) c = (21) verify (AB) C = A (BC).…
- If a = (ll 5&2 7&3) b = (rr 3&-1 -1&1) verify that (AB)T = BT AT.…
- Prove that a = (ll 5&2 7&3) b = (rr 3&-2 -7&5) are inverses to each other under…
- Solve (rr x&1) (rr 1&0 -2&-3) (x 5) = (0) .
- If a = (ll 3&3 7&6) , b = (ll 8&7 0&9) c = (rr 2&-3 4&6) , find (A + B)C and AC…

**Exercise 4.4**- Which one of the following statements is not true?A. A scalar matrix is a square…
- Matrix A-[aij]mxn is a square matrix ifA. m n B. m n C. m = 1 D. m = n…
- If (cc 3x+7&5 y+1&2-3x) = (cc 1 8&8) then the values of x and y respectively…
- If A = (1 -2 3) and b = (c -1 2 -3) then A + BA. (0 0 0) B. (0 0 0) C. (-14) D.…
- If a matrix is of order 2 × 3, then the number of elements in the matrix isA. 5…
- If (ll 8&4 x&8) = 4 (ll 2&1 1&2) then the value of x isA. 1 B. 2 C. 1/4 D. 4…
- If A is of order 3 × 4 and B is of order 4 × 3, then the order of BA isA. 3 × 3…
- If a x (ll 1&1 0&2) = (1 , 2) , then the order of A isA. 2 × 1 B. 2 × 2 C. 1 × 2…
- If A and B are square matrices such that AB = I and BA = I, then B isA. Unit…
- If (ll 1&2 2&1) (x y) = (2 4) , then the values of x and y respectively, areA.…
- If A = (cc 1&-2 -3&4) and A + B = O, then B isA. (rr 1&-2 -3&4) B. (rr -1&2…
- If A = (cc 4&-2 6&-3) , then A^2 isA. (ll 16&4 36&9) B. (8-4 12-6) C. (rr -4&2…
- A is of order m x n and B is of order p x q, addition of A and B is possible…
- If (rr a&3 1&2) (r 2 -1) = (5 0) , then the value of a isA. 8 B. 4 C. 2 D. 11…
- If A = (ll alpha & beta gamma & - alpha) is such that A^2 = I, thenA. 1 + Î±^2 +…
- If A = [aij]2x2 and aij = i + j, then A =A. (12 34) B. (ll 2&3 3&4) C. (23 45)…
- (cc -1&0 0&1) (ll a c) = (cc 1&0 0&-1) , then the values of a, b, c and d…
- If a = (ll 7&2 1&3) a+b = (cc -1&0 2&-4) , then the matrix B =A. (ll 1&0 0&1)…
- If (ccc 5&1) (r 2 -1 3) = (20) , then the value of x isA. 7 B. -7 C. 1/7 D. 0…
- Which one of the following is true for any two square matrices A and B of same…

**Exercise 4.1**

- The rates for the entrance tickets at a water theme park are listed below: Write…
- There are 6 Higher Secondary Schools, 8 High Schools and 13 Primary Schools in a…
- Find the order of the following matrices. (i) (ccc 1&-1&5 -2&3&4) (ii) (7 8 9)…
- A matrix has 8 elements. What are the possible orders it can have?…
- Matrix consists of 30 elements. What are the possible orders it can have?…
- Construct a 2x2 matrix A=[aij] whose elements are given by (i) aij = ij (ii) aij…
- Construct a 3 x 2 matrix A=[aij] whose elements are given by (i) a_ij = i/j (ii)…
- If A = (lrrr 1&-1&3&2 5&-4&7&4 6&0&9&8) , (i) find the order of the matrix (ii)…
- If a = (ll 2&3 4&1 5&0) , then find the transpose of A.
- If a = (ccc 1&2&3 2&4&5 3&-5&6) , then verify that (aT)T= A.

**Exercise 4.2**

- Find the values of x, y and z from the matrix equation (cc 5x+2 0&4z+6) = (rr…
- Solve for x and y if (2x+y x-3y) = (r 5 13)
- If a = (rr 2&3 -9&5) - (rr 1&5 7&-1) , then find the additive inverse of A.…
- Let a = (ll 3&2 5&1) b = (cc 8&-1 4&3) . Find the matrix C if C = 2A + B.…
- If a = (c 4-2 5-9) b = (rr 8&2 -1&-3) find 6A - 3B.
- Find a and b if a (2 3) + b (r -1 1) = (c 10 5) .
- Find X and Y if 2X + 3Y = (ll 2&3 4&0) and 3X + 2Y = (rr 2&-2 -1&5) .…
- Solve for x and y if (x^2 y^2) + 3 (c 2x -y) = (r -9 4)
- if a = (ll 3&2 5&1) , b = (lr 1-2 2&3) o = (ll 0&0 0&0) then Verify: (i) A + B =…
- If then verify that A + (B + C) = (A + B) + C.
- An electronic company records each type of entertainment device sold at three…
- The fees structure for one - day admission to a swimming pool is as follows:…

**Exercise 4.3**

- Determine whether the product of the matrices is defined in each case. If so,…
- Find the product of the matrices, if exists, (i) (2-1) (5 4) (ii) (cc 3&-2 5&1)…
- A fruit vendor sells fruits from his shop. Selling prices of Apple, Mango and…
- Find the values of x and y if (ll 1&2 3&3) (ll x&0 0) = (ll x&0 9&0) .…
- If a = (ll 5&3 7&5) , x = (x y) c = (c -5 -11) and if AX = C, then find the…
- If a = (ll 1&-1 2&3) then show that A^2 = 4A + 5I2 = O.
- If a = (ll 3&2 4&0) b = (ll 3&0 3&2) then find AB and BA. Are they equal?…
- If a = (rrr -1&2&1 1&2&3) , b = (0 1 2) c = (21) verify (AB) C = A (BC).…
- If a = (ll 5&2 7&3) b = (rr 3&-1 -1&1) verify that (AB)T = BT AT.…
- Prove that a = (ll 5&2 7&3) b = (rr 3&-2 -7&5) are inverses to each other under…
- Solve (rr x&1) (rr 1&0 -2&-3) (x 5) = (0) .
- If a = (ll 3&3 7&6) , b = (ll 8&7 0&9) c = (rr 2&-3 4&6) , find (A + B)C and AC…

**Exercise 4.4**

- Which one of the following statements is not true?A. A scalar matrix is a square…
- Matrix A-[aij]mxn is a square matrix ifA. m n B. m n C. m = 1 D. m = n…
- If (cc 3x+7&5 y+1&2-3x) = (cc 1 8&8) then the values of x and y respectively…
- If A = (1 -2 3) and b = (c -1 2 -3) then A + BA. (0 0 0) B. (0 0 0) C. (-14) D.…
- If a matrix is of order 2 × 3, then the number of elements in the matrix isA. 5…
- If (ll 8&4 x&8) = 4 (ll 2&1 1&2) then the value of x isA. 1 B. 2 C. 1/4 D. 4…
- If A is of order 3 × 4 and B is of order 4 × 3, then the order of BA isA. 3 × 3…
- If a x (ll 1&1 0&2) = (1 , 2) , then the order of A isA. 2 × 1 B. 2 × 2 C. 1 × 2…
- If A and B are square matrices such that AB = I and BA = I, then B isA. Unit…
- If (ll 1&2 2&1) (x y) = (2 4) , then the values of x and y respectively, areA.…
- If A = (cc 1&-2 -3&4) and A + B = O, then B isA. (rr 1&-2 -3&4) B. (rr -1&2…
- If A = (cc 4&-2 6&-3) , then A^2 isA. (ll 16&4 36&9) B. (8-4 12-6) C. (rr -4&2…
- A is of order m x n and B is of order p x q, addition of A and B is possible…
- If (rr a&3 1&2) (r 2 -1) = (5 0) , then the value of a isA. 8 B. 4 C. 2 D. 11…
- If A = (ll alpha & beta gamma & - alpha) is such that A^2 = I, thenA. 1 + Î±^2 +…
- If A = [aij]2x2 and aij = i + j, then A =A. (12 34) B. (ll 2&3 3&4) C. (23 45)…
- (cc -1&0 0&1) (ll a c) = (cc 1&0 0&-1) , then the values of a, b, c and d…
- If a = (ll 7&2 1&3) a+b = (cc -1&0 2&-4) , then the matrix B =A. (ll 1&0 0&1)…
- If (ccc 5&1) (r 2 -1 3) = (20) , then the value of x isA. 7 B. -7 C. 1/7 D. 0…
- Which one of the following is true for any two square matrices A and B of same…

###### Exercise 4.1

**Question 1.**The rates for the entrance tickets at a water theme park are listed below:

Write down the matrices for the rates of entrance tickets for adults, children and senior citizens. Also find the dimensions of the matrices.

**Answer:**The given table can be expressed as a matrix where each column denotes the week and weekend rates for Adult, Children and Senior Citizen.

The dimension of above matrix is 3 × 2.

Also, the same information can also be expressed as a matrix where each row denotes the week and weekend rates for Adult, Children and Senior Citizen.

The dimension of above matrix is 2 × 3.

**Question 2.**There are 6 Higher Secondary Schools, 8 High Schools and 13 Primary Schools in a town. Represent these data in the form of 3 × 1 and 1 × 3 matrices.

**Answer:**Representing the given information in a 3 × 1 matrix, we get,

Also, representing the given information in a 1 × 3 matrix, we get,

⇒ (6 8 13)

**Question 3.**Find the order of the following matrices.

(i)

(ii)

(iii)

(iv) (3 4 5)

(v)

**Answer:**(i) Order is 2 × 3

(ii) Order is 3 × 1

(iii) Order is 3 × 3

(iv) Order is 1 × 3

(v) Order is 4 × 2

**Question 4.**A matrix has 8 elements. What are the possible orders it can have?

**Answer:**Since there are 8 elements, we can make the multiples of 8, which are:- 1, 8, 2, 4

Therefore, the possible orders of a matrix are 1×8; 8×1; 2×4 and 4×2.

**Question 5.**Matrix consists of 30 elements. What are the possible orders it can have?

**Answer:**Since there are 30 elements, we can make the multiples of 30, which are:- 1×30; 30×1; 2×15; 15×2; 3×10; 10×3; 5×6 and 6×5

Therefore, the possible orders of a matrix are 1×30; 30×1; 2×15; 15×2; 3×10; 10×3; 5×6 and 6×5

**Question 6.**Construct a 2x2 matrix A=[a_{ij}] whose elements are given by

(i) a_{ij} = ij

(ii) a_{ij} = 2i – j

(iii)

**Answer:**(i) Since a_{ij} = i × j, and the general of matrix is:

On substituting the values, we get,

(ii) Since the general of matrix is:

On substituting the values, we get,

(iii) Since the general of matrix is:

On substituting the values, we get,

**Question 7.**Construct a 3 x 2 matrix A=[a_{ij}] whose elements are given by

(i)

(ii)

(iii)

**Answer:**(i) Since the general of matrix is:

On substituting the values, we get,

(ii) Since the general of matrix is:

On substituting the values, we get,

(iii) Since the general of matrix is:

On substituting the values, we get,

**Question 8.**If A = , (i) find the order of the matrix

(ii) write down the elements A_{24} and a_{32}

(iii) in which row and column does the element 7 occur?

**Answer:**(i) Order of matrix is 3 × 4.

(ii) Since the general of matrix is:

⇒ a_{24} = 4 and a_{32} = 0

(iii) Element 7 occurs in 2^{nd} row 3^{rd} column

**Question 9.**If , then find the transpose of *A*.

**Answer:**For the transpose of a matrix, we know that,

A^{T}_{ij} = A_{ji}, therefore the transpose of matrix A is:-

**Question 10.**If , then verify that (a^{T})^{T}= A.

**Answer:**For the transpose of a matrix, we know that,

A^{T}_{ij} = A_{ji}, therefore the transpose of matrix A is:-

Now, applying transpose on A^{T} , we get,

∴ We see that (A^{T})^{T} = A

**Question 1.**

The rates for the entrance tickets at a water theme park are listed below:

Write down the matrices for the rates of entrance tickets for adults, children and senior citizens. Also find the dimensions of the matrices.

**Answer:**

The given table can be expressed as a matrix where each column denotes the week and weekend rates for Adult, Children and Senior Citizen.

The dimension of above matrix is 3 × 2.

Also, the same information can also be expressed as a matrix where each row denotes the week and weekend rates for Adult, Children and Senior Citizen.

The dimension of above matrix is 2 × 3.

**Question 2.**

There are 6 Higher Secondary Schools, 8 High Schools and 13 Primary Schools in a town. Represent these data in the form of 3 × 1 and 1 × 3 matrices.

**Answer:**

Representing the given information in a 3 × 1 matrix, we get,

Also, representing the given information in a 1 × 3 matrix, we get,

⇒ (6 8 13)

**Question 3.**

Find the order of the following matrices.

(i)

(ii)

(iii)

(iv) (3 4 5)

(v)

**Answer:**

(i) Order is 2 × 3

(ii) Order is 3 × 1

(iii) Order is 3 × 3

(iv) Order is 1 × 3

(v) Order is 4 × 2

**Question 4.**

A matrix has 8 elements. What are the possible orders it can have?

**Answer:**

Since there are 8 elements, we can make the multiples of 8, which are:- 1, 8, 2, 4

Therefore, the possible orders of a matrix are 1×8; 8×1; 2×4 and 4×2.

**Question 5.**

Matrix consists of 30 elements. What are the possible orders it can have?

**Answer:**

Since there are 30 elements, we can make the multiples of 30, which are:- 1×30; 30×1; 2×15; 15×2; 3×10; 10×3; 5×6 and 6×5

Therefore, the possible orders of a matrix are 1×30; 30×1; 2×15; 15×2; 3×10; 10×3; 5×6 and 6×5

**Question 6.**

Construct a 2x2 matrix A=[a_{ij}] whose elements are given by

(i) a_{ij} = ij

(ii) a_{ij} = 2i – j

(iii)

**Answer:**

(i) Since a_{ij} = i × j, and the general of matrix is:

On substituting the values, we get,

(ii) Since the general of matrix is:

On substituting the values, we get,

(iii) Since the general of matrix is:

On substituting the values, we get,

**Question 7.**

Construct a 3 x 2 matrix A=[a_{ij}] whose elements are given by

(i)

(ii)

(iii)

**Answer:**

(i) Since the general of matrix is:

On substituting the values, we get,

(ii) Since the general of matrix is:

On substituting the values, we get,

(iii) Since the general of matrix is:

On substituting the values, we get,

**Question 8.**

If A = , (i) find the order of the matrix

(ii) write down the elements A_{24} and a_{32}

(iii) in which row and column does the element 7 occur?

**Answer:**

(i) Order of matrix is 3 × 4.

(ii) Since the general of matrix is:

⇒ a_{24} = 4 and a_{32} = 0

(iii) Element 7 occurs in 2^{nd} row 3^{rd} column

**Question 9.**

If , then find the transpose of *A*.

**Answer:**

For the transpose of a matrix, we know that,

A^{T}_{ij} = A_{ji}, therefore the transpose of matrix A is:-

**Question 10.**

If , then verify that (a^{T})^{T}= A.

**Answer:**

For the transpose of a matrix, we know that,

A^{T}_{ij} = A_{ji}, therefore the transpose of matrix A is:-

Now, applying transpose on A^{T} , we get,

∴ We see that (A^{T})^{T} = A

###### Exercise 4.2

**Question 1.**Find the values of x, y and z from the matrix equation

**Answer:**Since given is the matrix equation we would equate the right hand side elements with left hand side elements

⇒ 5x + 2 = 12 , y – 4 = – 8 and 4z + 6 = 2

⇒ 5x = 10 , y = – 4 and 4z = – 4

⇒ x = 2 , y = – 4 and z = – 1

**Question 2.**Solve for x and y if

**Answer:**Since given is the matrix equation we would equate the right hand side elements with left hand side elements

2x + y = 5 …1

and x – 3y = 13 ….2

Multiplying equation 2 by 2

we get (x – 3y = 13 ) 2

2x – 6y = 26 ….3

Subtracting equation 1 and 3

Or

Y = – 3

Substituting value of y in 1

2x – 3 = 5 ⇒ 2x = 8 ⇒ x = 4

Hence the solution is x = 4 and y = – 3

**Question 3.**If , then find the additive inverse of A.

**Answer:**Let us first solve for the value of matrix A =

⇒ A =

The additive inverse of A = negative of the matrix

– A = additive inverse of A =

**Question 4.**Let . Find the matrix C if C = 2A + B.

**Answer:**Given A = and B =

We have to find matrix C where C = 2A + B

Now 2A = 2

We would multiply each term of A with 2

2A =

2A + B = =

C =

**Question 5.**If find 6A – 3B.

**Answer:**Given A =

B =

6A =

3B = 3

6A – 3B =

6A – 3B =

**Question 6.**Find a and b if a .

**Answer:**

⇒

⇒

Equating both the sides of the matrix equation

We get

2a –b = 10 …..1

And 3a + b = 5 …..2

Adding equation 1 and 2

⇒ a = 3

Putting value of a in 1

2a –b = 10

⇒ 6 –b = 10 ⇒ b = – 4

Thus the required solutions are a = 3 and b = – 4

**Question 7.**Find X and Y if 2X + 3Y = and 3X + 2Y = .

**Answer:**Given 2x + 3y = ….1

and 3x + 2y = ...2

Adding 1 and 3 equations we get,

5x + 5y =

Dividing both the sides by 5

x + y = …3

Now subtracting 1 from 2

X – y = ……4

Adding 3 and 4

2x =

X =

Subtracting 4 from 3

2y =

Y =

Hence x = = and y =

**Question 8.**Solve for x and y if

**Answer:**given

⇒

⇒

Equating each element of the matrix equation with corresponding element

x^{2} + 6x = – 9

⇒ x^{2} + 6x + 9 = 0

⇒ x( x + 3) + 3(x + 3) = 0

⇒ (x + 3) (x + 3) = 0

x = – 3 , – 3

y^{2} – 3y = 4

⇒ y^{2} – 3y – 4 = 0

⇒ y( y – 4) + 1( y – 4) = 0

⇒ (y + 1) (y – 4) = 0

⇒ y = – 1 or 4

Hence the values of x = – 3, – 3 and y = – 1 , 4

**Question 9.**if then

Verify: (i) A + B = B + A (ii) A + ( – A) = O = ( – A) + A.

**Answer:**9) given

A =

(i) To verify A + B = B + A

LHS:

A + B =

RHS = B + A

=

Here, LHS = RHS hence proved

(ii) A + ( – A) = 0 = ( – A) + A

– A =

LHS:

= A + (– A)

=

RHS ( – A) + A =

Hence LHS = RHS = 0 proved

**Question 10.**If then

verify that A + (B + C) = (A + B) + C.

**Answer:**10) given

A = , B = and C

LHS = A + (B + C)

⇒

⇒

⇒

RHS = (A + B) + C

⇒

⇒

⇒

LHS = RHS

Hence proved

**Question 11.**An electronic company records each type of entertainment device sold at three of their branch stores so that they can monitor their purchases of supplies. The sales in two weeks are shown in the following spreadsheets.

Find the sum of the items sold out in two weeks using matrix addition.

**Answer:**Here we consider the types of items sold along the column and the store in which they are stored along the rows

Thus week 1 and week 2 matrices can be written as

W_{1} = and W_{2} =

The sum of the items sold in two weeks is the sum of the above two matrices, which is sum of each corresponding elements of the two matrices

W_{1} + W_{2} =

⇒

TV DVD video CD

⇒ The sum of items sold in two weeks =

**Question 12.**The fees structure for one – day admission to a swimming pool is as follows:

Write the matrix that represents the additional cost for non – membership.

**Answer:**Here we consider the type of member along the columns and the timings in the rows. Thus members and non – members matrices can be written as:

M = and N = respectively

The additional cost for non – members as compared to the members is the difference of the above two matrices, which is the difference of each element of the matrices to its corresponding element in the other matrix

N –M =

⇒

The additional cost of non – members as compared to the members is

**Question 1.**

Find the values of x, y and z from the matrix equation

**Answer:**

Since given is the matrix equation we would equate the right hand side elements with left hand side elements

⇒ 5x + 2 = 12 , y – 4 = – 8 and 4z + 6 = 2

⇒ 5x = 10 , y = – 4 and 4z = – 4

⇒ x = 2 , y = – 4 and z = – 1

**Question 2.**

Solve for x and y if

**Answer:**

Since given is the matrix equation we would equate the right hand side elements with left hand side elements

2x + y = 5 …1

and x – 3y = 13 ….2

Multiplying equation 2 by 2

we get (x – 3y = 13 ) 2

2x – 6y = 26 ….3

Subtracting equation 1 and 3

Or

Y = – 3

Substituting value of y in 1

2x – 3 = 5 ⇒ 2x = 8 ⇒ x = 4

Hence the solution is x = 4 and y = – 3

**Question 3.**

If , then find the additive inverse of A.

**Answer:**

Let us first solve for the value of matrix A =

⇒ A =

The additive inverse of A = negative of the matrix

– A = additive inverse of A =

**Question 4.**

Let . Find the matrix C if C = 2A + B.

**Answer:**

Given A = and B =

We have to find matrix C where C = 2A + B

Now 2A = 2

We would multiply each term of A with 2

2A =

2A + B = =

C =

**Question 5.**

If find 6A – 3B.

**Answer:**

Given A =

B =

6A =

3B = 3

6A – 3B =

6A – 3B =

**Question 6.**

Find a and b if a .

**Answer:**

⇒

⇒

Equating both the sides of the matrix equation

We get

2a –b = 10 …..1

And 3a + b = 5 …..2

Adding equation 1 and 2

⇒ a = 3

Putting value of a in 1

2a –b = 10

⇒ 6 –b = 10 ⇒ b = – 4

Thus the required solutions are a = 3 and b = – 4

**Question 7.**

Find X and Y if 2X + 3Y = and 3X + 2Y = .

**Answer:**

Given 2x + 3y = ….1

and 3x + 2y = ...2

Adding 1 and 3 equations we get,

5x + 5y =

Dividing both the sides by 5

x + y = …3

Now subtracting 1 from 2

X – y = ……4

Adding 3 and 4

2x =

X =

Subtracting 4 from 3

2y =

Y =

Hence x = = and y =

**Question 8.**

Solve for x and y if

**Answer:**

given

⇒

⇒

Equating each element of the matrix equation with corresponding element

x^{2} + 6x = – 9

⇒ x^{2} + 6x + 9 = 0

⇒ x( x + 3) + 3(x + 3) = 0

⇒ (x + 3) (x + 3) = 0

x = – 3 , – 3

y^{2} – 3y = 4

⇒ y^{2} – 3y – 4 = 0

⇒ y( y – 4) + 1( y – 4) = 0

⇒ (y + 1) (y – 4) = 0

⇒ y = – 1 or 4

Hence the values of x = – 3, – 3 and y = – 1 , 4

**Question 9.**

if then

Verify: (i) A + B = B + A (ii) A + ( – A) = O = ( – A) + A.

**Answer:**

9) given

A =

(i) To verify A + B = B + A

LHS:

A + B =

RHS = B + A

=

Here, LHS = RHS hence proved

(ii) A + ( – A) = 0 = ( – A) + A

– A =

LHS:

= A + (– A)

=

RHS ( – A) + A =

Hence LHS = RHS = 0 proved

**Question 10.**

If then

verify that A + (B + C) = (A + B) + C.

**Answer:**

10) given

A = , B = and C

LHS = A + (B + C)

⇒

⇒

⇒

RHS = (A + B) + C

⇒

⇒

⇒

LHS = RHS

Hence proved

**Question 11.**

An electronic company records each type of entertainment device sold at three of their branch stores so that they can monitor their purchases of supplies. The sales in two weeks are shown in the following spreadsheets.

Find the sum of the items sold out in two weeks using matrix addition.

**Answer:**

Here we consider the types of items sold along the column and the store in which they are stored along the rows

Thus week 1 and week 2 matrices can be written as

W_{1} = and W_{2} =

The sum of the items sold in two weeks is the sum of the above two matrices, which is sum of each corresponding elements of the two matrices

W_{1} + W_{2} =

⇒

TV DVD video CD

⇒ The sum of items sold in two weeks =

**Question 12.**

The fees structure for one – day admission to a swimming pool is as follows:

Write the matrix that represents the additional cost for non – membership.

**Answer:**

Here we consider the type of member along the columns and the timings in the rows. Thus members and non – members matrices can be written as:

M = and N = respectively

The additional cost for non – members as compared to the members is the difference of the above two matrices, which is the difference of each element of the matrices to its corresponding element in the other matrix

N –M =

⇒

The additional cost of non – members as compared to the members is

###### Exercise 4.3

**Question 1.**Determine whether the product of the matrices is defined in each case. If so, state the

order of the product.

(i) AB, where A = [a_{ij}]_{4x3}, B = [b_{ij}]_{3x2}

(ii)PQ, where P = [p_{ij}]_{4x3}, Q = [q_{ij}]_{4x3}

(iii)MN, where M = [m_{ij}]_{3x1}, N = [n_{ij}]_{1x5}

(iv) RS, where R = [r_{ij}]_{2x2}, S = [s_{ij}]_{2x2}

**Answer:**(i) The multiplication of 2 matrices is possible if number of columns in first matrix is equal to number of rows in second.

⇒ Here A[a_{ij}]_{4 x 3} and B = [b_{ij}]_{3x2}

⇒ Number of columns in A = 3

⇒ Number of rows in B = 3

Thus the product is defined and the order if product is

Number of rows in A × Number of columns in B

∴ AB = 4 × 3

(ii) The multiplication of 2 matrices is possible if number of columns in first matrix is equal to number of rows in second.

⇒ Here P[p_{ij}]_{4 x 3} and Q = [q_{ij}]_{4x3}

⇒ Number of columns in P = 3

⇒ Number of rows in Q = 4

Thus the product is not defined.

(iii) The multiplication of 2 matrices is possible if number of columns in first matrix is equal to number of rows in second.

⇒ Here M[m_{ij}]_{3 x 1} and N = [n_{ij}]_{1x5}

⇒ Number of columns in M = 1

⇒ Number of rows in N = 1

Thus the product is defined and the order if product is

Number of rows in M × Number of columns in N

∴ MN = 3 × 5

(iv) The multiplication of 2 matrices is possible if number of columns in first matrix is equal to number of rows in second.

⇒ Here R[r_{ij}]_{2 x 2} and S = [s_{ij}]_{2x2}

⇒ Number of columns in R = 2

⇒ Number of rows in S = 2

Thus the product is defined and the order if product is

Number of rows in R × Number of columns in S

∴ RS = 2 × 2

**Question 2.**Find the product of the matrices, if exists,

(i) (ii)

(iii) (iv)

**Answer:**(i) ⇒ let A : [2 -1] ∴ A[a_{ij}]_{1 × 2}

⇒ let B : ∴ B[b_{ij}]_{2 × 1}

Number of columns in A = 2

Number of rows in B = 2

Thus the product is defined and the order if product is

Number of rows in A × Number of columns in B

∴ AB = 1 × 1

⇒ [2 × 5 + (-1) × 4]

⇒ [10-4]

⇒ [6]

(ii) ⇒ let A : ∴ A[a_{ij}]_{2 × 2}

⇒ let B : ∴ B[b_{ij}]_{2 × 2}

Number of columns in A = 2

Number of rows in B = 2

Thus the product is defined and the order if product is

Number of rows in A × Number of columns in B

∴ AB = 2 × 2

⇒

⇒

(iii) ⇒ let A : ∴ A[a_{ij}]_{2 × 3}

⇒ let B : ∴ B[b_{ij}]_{3 × 2}

Number of columns in A = 3

Number of rows in B = 3

Thus the product is defined and the order if product is

Number of rows in A × Number of columns in B

∴ AB = 2 × 2

⇒

⇒

(iv) let A : ∴ A[a_{ij}]_{2 × 1}

let B : [2 7] ∴ B[b_{ij}]_{1 × 2}

Number of columns in A = 1

Number of rows in B = 1

Thus the product is defined and the order if product is

Number of rows in A × Number of columns in B

∴ AB = 2 × 2

× [2 -7]

⇒

⇒

**Question 3.**A fruit vendor sells fruits from his shop. Selling prices of Apple, Mango and Orange are ₹20, ₹10 and ₹5 each respectively. The sales in three days are given below

Write the matrix indicating the total amount collected on each day and hence find the total amount collected from selling of all three fruits combined.

**Answer:**⇒ Let the Sales matrix be A =

⇒ Selling price matrix B =

AB =

=

=

These are the amounts earned on each day.

Total amount earned = 1750 + 1600 + 1650 = 5000 Rs

**Question 4.**Find the values of x and y if .

**Answer:**=

=

=

Comparing with

3x = 9

∴ x = 3

And Y = 0

**Question 5.**If and if AX = C, then find the values of x and y.

**Answer:**x = 2, y = - 5

⇒ AX =

=

Comparing with

5x + 3y = -5 ---1

7x + 5y = -11 ---2

Multiply 1 by 5 and multiply 2 by 3 and subtract,

x = 2 and y = -5

**Question 6.**If then show that A^{2} = 4A + 5I_{2} = O.

**Answer:**⇒ A^{2} = AA =

=

=

⇒ 4A =

=

⇒ 5I =

⇒ A^{2} - 4A + 5I_{2} =

=

=

**Question 7.**If then find AB and BA. Are they equal?

**Answer:**⇒ AB =

=

=

⇒ BA =

=

=

AB not equal to BA

**Question 8.**If verify (AB) C = A (BC).

**Answer:**⇒ AB =

=

=

⇒ (AB)C =

=

=

⇒ BC =

=

=

⇒ A(BC) =

=

=

⇒ (AB) C = A (BC)

Thus verified.

**Question 9.**If verify that (AB)^{T} = B^{T} A^{T}.

**Answer:**⇒ AB =

=

=

⇒ (AB)^{T} =

⇒ B^{T} =

⇒ A^{T} =

⇒ B^{T} A^{T} =

=

=

⇒ (AB)^{T} = B^{T} A^{T}

This proved.

**Question 10.**Prove that are inverses to each other under matrix multiplication.

**Answer:**⇒ AB =

=

=

⇒ BA =

=

=

Thus A and B are inverse to each other

**Question 11.**Solve .

**Answer:**Let A = [x 1] B = C =

⇒ BC =

=

=

⇒ A(BC) = [x 1]

=

= = 0

∴ x = -3, x = 5

**Question 12.**If , find (A + B)C and AC + BCIs (A + B)C = AC + BC ?

**Answer:**⇒ A = , B = and C =

⇒ A + B =

=

⇒ A + B =

⇒ (A + B)C =

=

=

⇒ AC =

=

=

⇒ BC =

=

=

⇒ AC + BC =

=

⇒ Thus (A + B)C = AC + BC is true

**Question 1.**

Determine whether the product of the matrices is defined in each case. If so, state the

order of the product.

(i) AB, where A = [a_{ij}]_{4x3}, B = [b_{ij}]_{3x2}

(ii)PQ, where P = [p_{ij}]_{4x3}, Q = [q_{ij}]_{4x3}

(iii)MN, where M = [m_{ij}]_{3x1}, N = [n_{ij}]_{1x5}

(iv) RS, where R = [r_{ij}]_{2x2}, S = [s_{ij}]_{2x2}

**Answer:**

(i) The multiplication of 2 matrices is possible if number of columns in first matrix is equal to number of rows in second.

⇒ Here A[a_{ij}]_{4 x 3} and B = [b_{ij}]_{3x2}

⇒ Number of columns in A = 3

⇒ Number of rows in B = 3

Thus the product is defined and the order if product is

Number of rows in A × Number of columns in B

∴ AB = 4 × 3

(ii) The multiplication of 2 matrices is possible if number of columns in first matrix is equal to number of rows in second.

⇒ Here P[p_{ij}]_{4 x 3} and Q = [q_{ij}]_{4x3}

⇒ Number of columns in P = 3

⇒ Number of rows in Q = 4

Thus the product is not defined.

(iii) The multiplication of 2 matrices is possible if number of columns in first matrix is equal to number of rows in second.

⇒ Here M[m_{ij}]_{3 x 1} and N = [n_{ij}]_{1x5}

⇒ Number of columns in M = 1

⇒ Number of rows in N = 1

Thus the product is defined and the order if product is

Number of rows in M × Number of columns in N

∴ MN = 3 × 5

(iv) The multiplication of 2 matrices is possible if number of columns in first matrix is equal to number of rows in second.

⇒ Here R[r_{ij}]_{2 x 2} and S = [s_{ij}]_{2x2}

⇒ Number of columns in R = 2

⇒ Number of rows in S = 2

Thus the product is defined and the order if product is

Number of rows in R × Number of columns in S

∴ RS = 2 × 2

**Question 2.**

Find the product of the matrices, if exists,

(i) (ii)

(iii) (iv)

**Answer:**

(i) ⇒ let A : [2 -1] ∴ A[a_{ij}]_{1 × 2}

⇒ let B : ∴ B[b_{ij}]_{2 × 1}

Number of columns in A = 2

Number of rows in B = 2

Thus the product is defined and the order if product is

Number of rows in A × Number of columns in B

∴ AB = 1 × 1

⇒ [2 × 5 + (-1) × 4]

⇒ [10-4]

⇒ [6]

(ii) ⇒ let A : ∴ A[a_{ij}]_{2 × 2}

⇒ let B : ∴ B[b_{ij}]_{2 × 2}

Number of columns in A = 2

Number of rows in B = 2

Thus the product is defined and the order if product is

Number of rows in A × Number of columns in B

∴ AB = 2 × 2

⇒

⇒

(iii) ⇒ let A : ∴ A[a_{ij}]_{2 × 3}

⇒ let B : ∴ B[b_{ij}]_{3 × 2}

Number of columns in A = 3

Number of rows in B = 3

Thus the product is defined and the order if product is

Number of rows in A × Number of columns in B

∴ AB = 2 × 2

⇒

⇒

(iv) let A : ∴ A[a_{ij}]_{2 × 1}

let B : [2 7] ∴ B[b_{ij}]_{1 × 2}

Number of columns in A = 1

Number of rows in B = 1

Thus the product is defined and the order if product is

Number of rows in A × Number of columns in B

∴ AB = 2 × 2

× [2 -7]

⇒

⇒

**Question 3.**

A fruit vendor sells fruits from his shop. Selling prices of Apple, Mango and Orange are ₹20, ₹10 and ₹5 each respectively. The sales in three days are given below

Write the matrix indicating the total amount collected on each day and hence find the total amount collected from selling of all three fruits combined.

**Answer:**

⇒ Let the Sales matrix be A =

⇒ Selling price matrix B =

AB =

=

=

These are the amounts earned on each day.

Total amount earned = 1750 + 1600 + 1650 = 5000 Rs

**Question 4.**

Find the values of x and y if .

**Answer:**

=

=

=

Comparing with

3x = 9

∴ x = 3

And Y = 0

**Question 5.**

If and if AX = C, then find the values of x and y.

**Answer:**

x = 2, y = - 5

⇒ AX =

=

Comparing with

5x + 3y = -5 ---1

7x + 5y = -11 ---2

Multiply 1 by 5 and multiply 2 by 3 and subtract,

x = 2 and y = -5

**Question 6.**

If then show that A^{2} = 4A + 5I_{2} = O.

**Answer:**

⇒ A^{2} = AA =

=

=

⇒ 4A =

=

⇒ 5I =

⇒ A^{2} - 4A + 5I_{2} =

=

=

**Question 7.**

If then find AB and BA. Are they equal?

**Answer:**

⇒ AB =

=

=

⇒ BA =

=

=

AB not equal to BA

**Question 8.**

If verify (AB) C = A (BC).

**Answer:**

⇒ AB =

=

=

⇒ (AB)C =

=

=

⇒ BC =

=

=

⇒ A(BC) =

=

=

⇒ (AB) C = A (BC)

Thus verified.

**Question 9.**

If verify that (AB)^{T} = B^{T} A^{T}.

**Answer:**

⇒ AB =

=

=

⇒ (AB)^{T} =

⇒ B^{T} =

⇒ A^{T} =

⇒ B^{T} A^{T} =

=

=

⇒ (AB)^{T} = B^{T} A^{T}

This proved.

**Question 10.**

Prove that are inverses to each other under matrix multiplication.

**Answer:**

⇒ AB =

=

=

⇒ BA =

=

=

Thus A and B are inverse to each other

**Question 11.**

Solve .

**Answer:**

Let A = [x 1] B = C =

⇒ BC =

=

=

⇒ A(BC) = [x 1]

=

= = 0

∴ x = -3, x = 5

**Question 12.**

If , find (A + B)C and AC + BCIs (A + B)C = AC + BC ?

**Answer:**

⇒ A = , B = and C =

⇒ A + B =

=

⇒ A + B =

⇒ (A + B)C =

=

=

⇒ AC =

=

=

⇒ BC =

=

=

⇒ AC + BC =

=

⇒ Thus (A + B)C = AC + BC is true

###### Exercise 4.4

**Question 1.**Which one of the following statements is not true?

A. A scalar matrix is a square matrix

B. A diagonal matrix is a square matrix

C. A scalar matrix is a diagonal matrix

D. A diagonal matrix is a scalar matrix.

**Answer:**In the above question we see the following terms,

Scalar Matrix, Square Matrix, Diagonal Matrix

To answer this question, we have to know the definition of the above terms.

__Square Matrix:__

If the rows and columns of the matrices are equal then it will constitute square like structure. So, it is called as square matrix.

E.g.

__Diagonal Matrix:__

In a square matrix all the elements are zero except the diagonal elements of the matrix. Then that matrix is said to be diagonal matrix.

E.g.

__Scalar Matrix:__

In a diagonal matrix, if all the diagonal elements are same then that matrix is said to be a scalar matrix.

E.g.

Option(A):

A scalar matrix should be a square matrix so, it is true

Option(B):

Diagonal matrix forms only with the square matrix so, it is true

Option(C):

A scalar matrix consists of zero except the diagonal elements so it is true.

Option(D):

A scalar matrix should comprise of same diagonal elements but in diagonal matrix it may (or) may not contains same diagonal elements. So it is False.

__Option(D) is not True in the given.__

**Question 2.**Matrix A-[a_{ij}]_{mxn} is a square matrix if

A. m < n

B. m > n

C. m = 1

D. m = n

**Answer:**Given that A is a matrix with m and n as their rows and columns respectively.

We know that for a square matrix both m and n should be equal

So, m = n is the correct answer.

**Question 3.**If then the values of x and y respectively are

A. –2, 7

B.

C.

D. 2, –7

**Answer:**Given,

=

If the one matrix is equivalent to other matrix, then their elements should be equal.

⇒ 3x + 7 = 1 & 5 = y-2

3x = -6 ; y = 7

x = -2 ; y = 7

∴ Option (A) is the answer.

**Question 4.**If A = (1 -2 3) and then A + B

A. (0 0 0)

B.

C. (-14)

D. not defined

**Answer:**For matrix addition operation their no. of rows and no. of columns should be equal. Otherwise addition is not possible.

In the given matrix A has 1 row and 3 columns

But matrix B has 3 rows and 1 column.

Since rows and columns are not equal it is not possible to add.

So the answer is (d)

**Question 5.**If a matrix is of order 2 × 3, then the number of elements in the matrix is

A. 5

B. 6

C. 2

D. 3

**Answer:**Given that a matrix with order 2x3

We know that the no. of elements in the matrix is equal to the product of no. of rows and no. of columns

i.e., No. of elements = No. of rows × No. of columns

∴ No. of elements = 2 x 3

= 6

∴ There will be 6 elements in the given matrix

So, option (B) is the correct answer.

**Question 6.**If then the value of x is

A. 1

B. 2

C.

D. 4

**Answer:**Given,

Since both the matrix is equal to each other, the elements in it also be equal.

By equalizing we will get x = 4

So, option (D) is correct answer.

**Question 7.**If A is of order 3 × 4 and B is of order 4 × 3, then the order of BA is

A. 3 × 3

B. 4 × 4

C. 4 × 3

D. not defined

**Answer:**Given matrix orders

[A] with 3 x 4 and [B] with 4 x 3

We have to find the order of [B].[A]

For matrix multiplication the columns of 1^{st} matrix and the rows of 2^{nd} matrix should be equal.

E.g. [A]_{3x2}.[B]_{2x2}

The order of new matrix formed after multiplication of matrices will be 1^{st} matrix row will be taken as new matrix rows. The no. of columns in 2^{nd} matrix will be taken as new matrix columns.

E.g. [A]_{3x2}.[B]_{2x2} = [AB]_{3x2}

In the same way [B]_{4x3}.[A]_{3x4} = [AB]_{4x4}

∴ option (B) is correct answer

**Question 8.**If , then the order of A is

A. 2 × 1

B. 2 × 2

C. 1 × 2

D. 3 × 2

**Answer:**Given,

A x

Let us consider as matrix B, it has an order of 2x2

Let us consider the resultant matrix as C, it has an order of 1x2

For matrix multiplication the columns of 1^{st} matrix and the rows of 2^{nd} matrix should be equal.

So, matrix A has got same Columns as Rows of matrix B

The order of new matrix formed after multiplication of matrices will be 1^{st} matrix row will be taken as new matrix rows. The no. of columns in 2^{nd} matrix will be taken as new matrix columns

So, matrix A has got same Rows as Rows of matrix C

So, Matrix A will consists of 1x2 order

∴ option (C) is correct answer

**Question 9.**If A and B are square matrices such that AB = I and BA = I, then B is

A. Unit matrix

B. Null matrix

C. Multiplicative inverse matrix of A

D. –A

**Answer:**It is based on a property about multiplication inverse. Only product of a matrix and its own inverse matrix then the resultant matrix will be identity matrix.

AB = I and BA = I

So, B is the multiplicative inverse matrix of A

∴ option (C) is correct answer.

**Question 10.**If , then the values of x and y respectively, are

A. 2, 0

B. 0, 2

C. 0, -2

D. 1, 1

**Answer:**Given, =

=

By equating all the elements in the matrix

We will get 2 equations

x + 2y = 2 ⇒ __1__

2x + y = 4 ⇒ __2__

By solving the both equations we will get

x = 2; y = 0

∴ option (A) is correct answer

**Question 11.**If A = and A + B = O, then B is

A.

B.

C.

D.

**Answer:**Given that

A + B = O ⇒ B = -A

∴ B =

B =

∴ option (B) is correct answer

**Question 12.**If A = , then A^{2} is

A.

B.

C.

D.

**Answer:**Given A =

A^{2} = A×A = x

=

=

=

∴ option (D) is correct answer

**Question 13.**A is of order m x n and B is of order p x q, addition of A and B is possible only if

A. m = p

B. n = q

C. n = p

D. m = p, n = q

**Answer:**Given matrix of A is m x n ⇒ [A]_{mxn}

Matrix of B is p x q ⇒ [B]_{pxq}

We know that addition is possible is possible only the order is same for both the matrices

So, m = p, n = q

So, option (D) is true

**Question 14.**If , then the value of a is

A. 8

B. 4

C. 2

D. 11

**Answer:** =

By multiplying the matrices, we will get the value of ‘a’

=

By equating the elements in the matrices

2a-3 = 5

2a = 8

a = 4

So, option (B) is correct answer

**Question 15.**If A = is such that A^{2} = I, then

A. 1 + Î±^{2} + Î²Î³ = 0

B. 1 – Î±^{2} + Î²Î³ = 0

C. 1 – Î±^{2} – Î²Î³ = 0

D. 1 + Î±^{2} – Î²Î³ = 0

**Answer:**Given A =

A^{2} = I

I is the identity matrix

=

By comparing the elements in the matrix

⇒

∴ Option (C) is correct answer.

**Question 16.**If A = [a_{ij}]_{2x2} and a_{ij} = i + j, then A =

A.

B.

C.

D.

**Answer:**Given A = [a_{ij}]_{2x2} and a_{ij} = i + j

It means matrix A with order 2x2 and the elements in is given by a_{ij} = i + j

a_{11} = 1 + 1 = 2

a_{12} = 1 + 2 = 3

a_{21} = 2 + 1 = 3

a_{22} = 2 + 2 = 4

A = =

∴ Option (B) is correct answer.

**Question 17.**, then the values of a, b, c and d

respectively are

A. -1, 0, 0, - 1

B. 1, 0, 0, 1

C. -1,0,1,0

D. 1, 0, 0, 0

**Answer:**Given

By multiplying the matrices, we will get the values of a, b, c & d

By comparing the matrices on both sides

-a = 1; -b = 0; c = 0; d = -1

∴ Option (A) is correct answer.

**Question 18.**If , then the matrix B =

A.

B.

C.

D.

**Answer:**Given

A + B = ; A =

B =

B =

=

=

∴ Option (C) is correct answer.

**Question 19.**If , then the value of x is

A. 7

B. -7

C.

D. 0

**Answer:**Given

By equating the elements

13 – x = 20

x = -7

∴ Option (B) is correct answer.

**Question 20.**Which one of the following is true for any two square matrices A and B of same order?.

A. (AB)^{T} = A^{T}B^{T}

B. (A^{T}B)^{T} = A^{T}B^{T}

C. (AB)^{T} = BA

D. (AB)^{T} = B^{T}A^{T}

**Answer:**This is Reversal law for transpose of matrices

∴ (AB)^{T} = B^{T}A^{T}

∴ Option (D) is correct answer.

**Question 1.**

Which one of the following statements is not true?

A. A scalar matrix is a square matrix

B. A diagonal matrix is a square matrix

C. A scalar matrix is a diagonal matrix

D. A diagonal matrix is a scalar matrix.

**Answer:**

In the above question we see the following terms,

Scalar Matrix, Square Matrix, Diagonal Matrix

To answer this question, we have to know the definition of the above terms.

__Square Matrix:__

If the rows and columns of the matrices are equal then it will constitute square like structure. So, it is called as square matrix.

E.g.

__Diagonal Matrix:__

In a square matrix all the elements are zero except the diagonal elements of the matrix. Then that matrix is said to be diagonal matrix.

E.g.

__Scalar Matrix:__

In a diagonal matrix, if all the diagonal elements are same then that matrix is said to be a scalar matrix.

E.g.

Option(A):

A scalar matrix should be a square matrix so, it is true

Option(B):

Diagonal matrix forms only with the square matrix so, it is true

Option(C):

A scalar matrix consists of zero except the diagonal elements so it is true.

Option(D):

A scalar matrix should comprise of same diagonal elements but in diagonal matrix it may (or) may not contains same diagonal elements. So it is False.

__Option(D) is not True in the given.__

**Question 2.**

Matrix A-[a_{ij}]_{mxn} is a square matrix if

A. m < n

B. m > n

C. m = 1

D. m = n

**Answer:**

Given that A is a matrix with m and n as their rows and columns respectively.

We know that for a square matrix both m and n should be equal

So, m = n is the correct answer.

**Question 3.**

If then the values of x and y respectively are

A. –2, 7

B.

C.

D. 2, –7

**Answer:**

Given,

=

If the one matrix is equivalent to other matrix, then their elements should be equal.

⇒ 3x + 7 = 1 & 5 = y-2

3x = -6 ; y = 7

x = -2 ; y = 7

∴ Option (A) is the answer.

**Question 4.**

If A = (1 -2 3) and then A + B

A. (0 0 0)

B.

C. (-14)

D. not defined

**Answer:**

For matrix addition operation their no. of rows and no. of columns should be equal. Otherwise addition is not possible.

In the given matrix A has 1 row and 3 columns

But matrix B has 3 rows and 1 column.

Since rows and columns are not equal it is not possible to add.

So the answer is (d)

**Question 5.**

If a matrix is of order 2 × 3, then the number of elements in the matrix is

A. 5

B. 6

C. 2

D. 3

**Answer:**

Given that a matrix with order 2x3

We know that the no. of elements in the matrix is equal to the product of no. of rows and no. of columns

i.e., No. of elements = No. of rows × No. of columns

∴ No. of elements = 2 x 3

= 6

∴ There will be 6 elements in the given matrix

So, option (B) is the correct answer.

**Question 6.**

If then the value of x is

A. 1

B. 2

C.

D. 4

**Answer:**

Given,

Since both the matrix is equal to each other, the elements in it also be equal.

By equalizing we will get x = 4

So, option (D) is correct answer.

**Question 7.**

If A is of order 3 × 4 and B is of order 4 × 3, then the order of BA is

A. 3 × 3

B. 4 × 4

C. 4 × 3

D. not defined

**Answer:**

Given matrix orders

[A] with 3 x 4 and [B] with 4 x 3

We have to find the order of [B].[A]

For matrix multiplication the columns of 1^{st} matrix and the rows of 2^{nd} matrix should be equal.

E.g. [A]_{3x2}.[B]_{2x2}

The order of new matrix formed after multiplication of matrices will be 1^{st} matrix row will be taken as new matrix rows. The no. of columns in 2^{nd} matrix will be taken as new matrix columns.

E.g. [A]_{3x2}.[B]_{2x2} = [AB]_{3x2}

In the same way [B]_{4x3}.[A]_{3x4} = [AB]_{4x4}

∴ option (B) is correct answer

**Question 8.**

If , then the order of A is

A. 2 × 1

B. 2 × 2

C. 1 × 2

D. 3 × 2

**Answer:**

Given,

A x

Let us consider as matrix B, it has an order of 2x2

Let us consider the resultant matrix as C, it has an order of 1x2

For matrix multiplication the columns of 1^{st} matrix and the rows of 2^{nd} matrix should be equal.

So, matrix A has got same Columns as Rows of matrix B

The order of new matrix formed after multiplication of matrices will be 1^{st} matrix row will be taken as new matrix rows. The no. of columns in 2^{nd} matrix will be taken as new matrix columns

So, matrix A has got same Rows as Rows of matrix C

So, Matrix A will consists of 1x2 order

∴ option (C) is correct answer

**Question 9.**

If A and B are square matrices such that AB = I and BA = I, then B is

A. Unit matrix

B. Null matrix

C. Multiplicative inverse matrix of A

D. –A

**Answer:**

It is based on a property about multiplication inverse. Only product of a matrix and its own inverse matrix then the resultant matrix will be identity matrix.

AB = I and BA = I

So, B is the multiplicative inverse matrix of A

∴ option (C) is correct answer.

**Question 10.**

If , then the values of x and y respectively, are

A. 2, 0

B. 0, 2

C. 0, -2

D. 1, 1

**Answer:**

Given, =

=

By equating all the elements in the matrix

We will get 2 equations

x + 2y = 2 ⇒ __1__

2x + y = 4 ⇒ __2__

By solving the both equations we will get

x = 2; y = 0

∴ option (A) is correct answer

**Question 11.**

If A = and A + B = O, then B is

A.

B.

C.

D.

**Answer:**

Given that

A + B = O ⇒ B = -A

∴ B =

B =

∴ option (B) is correct answer

**Question 12.**

If A = , then A^{2} is

A.

B.

C.

D.

**Answer:**

Given A =

A^{2} = A×A = x

=

=

=

∴ option (D) is correct answer

**Question 13.**

A is of order m x n and B is of order p x q, addition of A and B is possible only if

A. m = p

B. n = q

C. n = p

D. m = p, n = q

**Answer:**

Given matrix of A is m x n ⇒ [A]_{mxn}

Matrix of B is p x q ⇒ [B]_{pxq}

We know that addition is possible is possible only the order is same for both the matrices

So, m = p, n = q

So, option (D) is true

**Question 14.**

If , then the value of a is

A. 8

B. 4

C. 2

D. 11

**Answer:**

=

By multiplying the matrices, we will get the value of ‘a’

=

By equating the elements in the matrices

2a-3 = 5

2a = 8

a = 4

So, option (B) is correct answer

**Question 15.**

If A = is such that A^{2} = I, then

A. 1 + Î±^{2} + Î²Î³ = 0

B. 1 – Î±^{2} + Î²Î³ = 0

C. 1 – Î±^{2} – Î²Î³ = 0

D. 1 + Î±^{2} – Î²Î³ = 0

**Answer:**

Given A =

A^{2} = I

I is the identity matrix

=

By comparing the elements in the matrix

⇒

∴ Option (C) is correct answer.

**Question 16.**

If A = [a_{ij}]_{2x2} and a_{ij} = i + j, then A =

A.

B.

C.

D.

**Answer:**

Given A = [a_{ij}]_{2x2} and a_{ij} = i + j

It means matrix A with order 2x2 and the elements in is given by a_{ij} = i + j

a_{11} = 1 + 1 = 2

a_{12} = 1 + 2 = 3

a_{21} = 2 + 1 = 3

a_{22} = 2 + 2 = 4

A = =

∴ Option (B) is correct answer.

**Question 17.**

, then the values of a, b, c and d

respectively are

A. -1, 0, 0, - 1

B. 1, 0, 0, 1

C. -1,0,1,0

D. 1, 0, 0, 0

**Answer:**

Given

By multiplying the matrices, we will get the values of a, b, c & d

By comparing the matrices on both sides

-a = 1; -b = 0; c = 0; d = -1

∴ Option (A) is correct answer.

**Question 18.**

If , then the matrix B =

A.

B.

C.

D.

**Answer:**

Given

A + B = ; A =

B =

B =

=

=

∴ Option (C) is correct answer.

**Question 19.**

If , then the value of x is

A. 7

B. -7

C.

D. 0

**Answer:**

Given

By equating the elements

13 – x = 20

x = -7

∴ Option (B) is correct answer.

**Question 20.**

Which one of the following is true for any two square matrices A and B of same order?.

A. (AB)^{T} = A^{T}B^{T}

B. (A^{T}B)^{T} = A^{T}B^{T}

C. (AB)^{T} = BA

D. (AB)^{T} = B^{T}A^{T}

**Answer:**

This is Reversal law for transpose of matrices

∴ (AB)^{T} = B^{T}A^{T}

∴ Option (D) is correct answer.