Question

If $A=\left[\begin{array}{ccc}1& 0& 1\\ 0& 2& 0\\ 1& -1& 4\end{array}\right],A=B+C,B=B^T$ and $C=-C^T$, then $C=$

• A. $\left[\begin{array}{ccc}0& 0.5& 0\\ -0.5& 0& 0\\ 0& 0& 0\end{array}\right]$
• B. $\left[\begin{array}{ccc}0& 0.5& 0\\ 0& 0& 0.5\\ 0& -0.5& 0\end{array}\right]$
• C. $\left[\begin{array}{ccc}0& -0.5& 0.5\\ 0.5& 0& 0\\ 0.5& 0& 0\end{array}\right]$
• D. $\left[\begin{array}{ccc}0& 0.5& 0\\ -0.5& 0& 0.5\\ 0& -0.5& 0\end{array}\right]$

Answer 1

Answer: (B)

$A=\left[\begin{array}{ccc}1& 0& 1\\ 0& 2& 0\\ 1& -1& 4\end{array}\right]$
$A^T=\left[\begin{array}{ccc}1& 0& 1\\ 0& 2& -1\\ 1& 0& 4\end{array}\right]$
$A+A^T=\left[\begin{array}{ccc}2& 0& 2\\ 0& 4& -1\\ 2& -1& 8\end{array}\right]A-A^T=\left[\begin{array}{ccc}0& 0& 0\\ 0& 0& 1\\ 0& -1& 0\end{array}\right]$
$A=\frac{1}{2}\left(A+A^T\right)+\frac{1}{2}\left(A-A^T\right)$
$A=\left[\begin{array}{ccc}1& 0& 1\\ 0& 2& -0.5\\ 1& -0.5& 4\end{array}\right]+\left[\begin{array}{ccc}0& 0& 0\\ 0& 0& 0.5\\ 0& -0.5& 0\end{array}\right]$
$A=B+C$
$C=\left[\begin{array}{ccc}0& 0& 0\\ 0& 0& 0.5\\ 0& -0.5& 0\end{array}\right]=-C^T$