Question

If a is $A$ symmetric matrix and $B$ is a skew-symmetrix matrix such that $A + B = \begin{bmatrix}2&3\\ 5&-1\end{bmatrix} $, then AB is equal to :

• A. $\begin{bmatrix} -4&2\\ 1&4\end{bmatrix}$
• B. $\begin{bmatrix} -4& -2\\ - 1&4\end{bmatrix}$
• C. $\begin{bmatrix} 4& -2\\ -1&-4\end{bmatrix}$
• D. $\begin{bmatrix} 4 & -2\\ 1 & -4\end{bmatrix}$

Answer 1

Answer: (C)

$A = A' , B = - B'$
$ A + B = \begin{bmatrix}2&3\\ 5&-1\end{bmatrix} $ ....(1)
$ A' + B' = \begin{bmatrix}2&5\\ 3 &-1\end{bmatrix} $
$ A -B = \begin{bmatrix}2&5\\ 3&-1\end{bmatrix} $ .....(2)
After adding Eq. (1) & (2)
$ A = \begin{bmatrix}2&4\\ 4 &-1\end{bmatrix} , B = \begin{bmatrix}0&-1\\ 1 &0\end{bmatrix} $
$ AB = \begin{bmatrix}4&-2\\ -1 &-4\end{bmatrix} $