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TS Inter 1st Year Maths 1A Matrices Solutions Exercise 3(c)

Answer: We have (AB)’ = B’A’ and (AB’)’ = (B’)’ A’ = BA’ (∵ (B )’ = B)

M Junaid Verified Source Updated: Jan 8, 2026 01:55 PM IST 1 min read

TS Inter 1st Year Maths 1A Matrices Solutions Exercise 3(c)

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Question 1.
If A = $\left[\begin{array}{rrr} 2 & 0 & 1 \\ -1 & 1 & 5 \end{array}\right]$ and B = $\left[\begin{array}{rrr} -1 & 1 & 0 \\ 0 & 1 & -2 \end{array}\right]$ then find (AB’)’

Answer:
We have (AB)’ = B’A’
and (AB’)’ = (B’)’ A’ = BA’ (∵ (B )’ = B)
$TS Inter 1st Year Maths 1A Solutions Chapter 3 Matrices Ex 3(c) 1$

Question 2.
If A = $\left[\begin{array}{rr} -2 & 1 \\ 5 & 0 \\ -1 & 4 \end{array}\right]$ and B = $\left[\begin{array}{rrr} -2 & 3 & 1 \\ 4 & 0 & 2 \end{array}\right]$ then find 2A + B’ and 3B’ – A.

Answer:
$TS Inter 1st Year Maths 1A Solutions Chapter 3 Matrices Ex 3(c) 2$

Question 3.
If A = $\left[\begin{array}{cc} 2 & -4 \\ -5 & 3 \end{array}\right]$ then find A + A’ and A. A’ (May 2007) (Board Model Paper)

Answer:
$TS Inter 1st Year Maths 1A Solutions Chapter 3 Matrices Ex 3(c) 3$

Question 4.
If A = $\left[\begin{array}{ccc} -1 & 2 & 3 \\ 2 & 5 & 6 \\ 3 & x & 7 \end{array}\right]$ is a symmetric matrix then find x.

Answer:
A matrix ‘A’ is said to be symmetric if A’ = A
$TS Inter 1st Year Maths 1A Solutions Chapter 3 Matrices Ex 3(c) 4$

Question 5.
If A = $\left[\begin{array}{ccc} 0 & 2 & 1 \\ -2 & 0 & -2 \\ -1 & x & 0 \end{array}\right]$ is a skew symmetric matrix, find x. (May 2014, 11)

Answer:
A matrix A is said to be skew symmetric if A’ = – A
$\left[\begin{array}{ccc} 0 & -2 & -1 \\ 2 & 0 & \mathrm{x} \\ 1 & -2 & 0 \end{array}\right]=\left[\begin{array}{ccc} 0 & -2 & -1 \\ 2 & 0 & 2 \\ 1 & -\mathrm{x} & 0 \end{array}\right]$
from equality of matrix x = 2

Question 6.
Is $\left[\begin{array}{ccc} 0 & 1 & 4 \\ -1 & 0 & 7 \\ -4 & -7 & 0 \end{array}\right]$ a symmetric or skew symmetric?

Answer:
Let A = $\left[\begin{array}{ccc} 0 & 1 & 4 \\ -1 & 0 & 7 \\ -4 & -7 & 0 \end{array}\right]$ then A is symmetric if A’ = A and skew symmetric if A’ = – A
i.e., A’ = $\left[\begin{array}{ccc} 0 & -1 & -4 \\ 1 & 0 & -7 \\ 4 & 7 & 0 \end{array}\right]=\left[\begin{array}{ccc} 0 & 1 & 4 \\ -1 & 0 & 7 \\ -4 & -7 & 0 \end{array}\right]$ = -A
∴ The matrix A is a skew symmetric matrix.

II.
Question 1.
If A = $\left[\begin{array}{cc} \cos \alpha & \sin \alpha \\ -\sin \alpha & \cos \alpha \end{array}\right]$ , show that A . A’ = A’ . A = I₂. (March 2007)

Answer:
$TS Inter 1st Year Maths 1A Solutions Chapter 3 Matrices Ex 3(c) 5$

Question 2.
If A = $\left[\begin{array}{ccc} 1 & 5 & 3 \\ 2 & 4 & 0 \\ 3 & -1 & -5 \end{array}\right]$ and B = $\left[\begin{array}{ccc} 2 & -1 & 0 \\ 0 & -2 & 5 \\ 1 & 2 & 0 \end{array}\right]$ , then find 3A – 4B’.

Answer:
$TS Inter 1st Year Maths 1A Solutions Chapter 3 Matrices Ex 3(c) 6$

Question 3.
If A = $\left[\begin{array}{rr} 7 & -2 \\ -1 & 2 \\ 5 & 3 \end{array}\right]$ and B = $\left[\begin{array}{rr} -2 & -1 \\ 4 & 2 \\ -1 & 0 \end{array}\right]$ then find AB’ and BA’.

Answer:
$TS Inter 1st Year Maths 1A Solutions Chapter 3 Matrices Ex 3(c) 7$

Question 4.
For any square matrix A; show that A A’ is symmetric. (March 2015-A.P)

Answer:
By definition a matrix is said to be symmetric if A’ = A.
∴(A A’)’ = (A’)’ A’ = A A’
[(∵ (AB)’ = B’A’ and (A’)’ = A]
Hence AA’ is a symmetric matrix.

TS Inter 1st Year Maths 1A Matrices Solutions Exercise 3(c)

Question 1. If A = [2−10115]\left[\begin{array}{rrr} 2 & 0 & 1 \\ -1 & 1 & 5 \end{array}\right] and B = [−10110−2]\left[\begin{array}{rrr} -1 & 1 & 0 \\ 0 & 1 & -2 \end{array}\right] then find (AB’)’

Question 2. If A = ⎡⎣⎢−25−1104⎤⎦⎥\left[\begin{array}{rr} -2 & 1 \\ 5 & 0 \\ -1 & 4 \end{array}\right] and B = [−243012]\left[\begin{array}{rrr} -2 & 3 & 1 \\ 4 & 0 & 2 \end{array}\right] then find 2A + B’ and 3B’ – A.

Question 3. If A = [2−5−43]\left[\begin{array}{cc} 2 & -4 \\ -5 & 3 \end{array}\right] then find A + A’ and A. A’ (May 2007) (Board Model Paper)

Question 4. If A = ⎡⎣⎢−12325x367⎤⎦⎥\left[\begin{array}{ccc} -1 & 2 & 3 \\ 2 & 5 & 6 \\ 3 & x & 7 \end{array}\right] is a symmetric matrix then find x.

Question 5. If A = ⎡⎣⎢0−2−120x1−20⎤⎦⎥\left[\begin{array}{ccc} 0 & 2 & 1 \\ -2 & 0 & -2 \\ -1 & x & 0 \end{array}\right] is a skew symmetric matrix, find x. (May 2014, 11)

Question 6. Is ⎡⎣⎢0−1−410−7470⎤⎦⎥\left[\begin{array}{ccc} 0 & 1 & 4 \\ -1 & 0 & 7 \\ -4 & -7 & 0 \end{array}\right] a symmetric or skew symmetric?

Question 2. If A = ⎡⎣⎢12354−130−5⎤⎦⎥\left[\begin{array}{ccc} 1 & 5 & 3 \\ 2 & 4 & 0 \\ 3 & -1 & -5 \end{array}\right] and B = ⎡⎣⎢201−1−22050⎤⎦⎥\left[\begin{array}{ccc} 2 & -1 & 0 \\ 0 & -2 & 5 \\ 1 & 2 & 0 \end{array}\right], then find 3A – 4B’.

Question 4. For any square matrix A; show that A A’ is symmetric. (March 2015-A.P)

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