Students must practice these TS Intermediate Maths 1A Solutions Chapter 3 Matrices Ex 3(c) to find a better approach to solving the problems.

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

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)

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:

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:

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

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 = I_{2}. (March 2007)

Answer:

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:

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:

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.