Question

if $A = \begin{bmatrix}3&-4\\ 1&-1\end{bmatrix}$ is the sum of a symmetric matrix $B$ and a skew-symmetric matrix $C$, then $C$ is

• A. $\begin{bmatrix}1&-\frac{5}{2}\\ \frac{5}{2}&0\end{bmatrix}$
• B. $\begin{bmatrix}1&-\frac{5}{2}\\ \frac{5}{2}&1\end{bmatrix}$
• C. $\begin{bmatrix}0&-\frac{5}{2}\\ \frac{5}{2}&0\end{bmatrix}$
• D. $\begin{bmatrix}0&-\frac{3}{2}\\ \frac{5}{2}&1\end{bmatrix}$

Answer 1

Answer: (C)

$A = \begin{bmatrix}3&-4\\ 1&-1\end{bmatrix} $
$A= \left(\frac{A + A'}{2}\right)+\left(\frac{A-A'}{2}\right) = B + C$
[where $B$ and $C$ are symmetric and skew-symmetric matrices respectively]
Now, $C = \frac{A - A'}{2} = \frac{1}{2 } \left\{\begin{bmatrix}3&-4\\ 1&-1\end{bmatrix} - \begin{bmatrix}3&1\\ -4&-1\end{bmatrix}\right\}$
$ = \frac{1}{2}\begin{bmatrix}0&-5\\ 5&0\end{bmatrix} = \begin{bmatrix}0&-\frac{5}{2}\\ \frac{5}{2}&0\end{bmatrix}$