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

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:

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.