Question

If the matrix$\left[\begin{array}{ccc}0& a& 3\\ 2& b& -1\\ c& 1& 0\end{array}\right]$is a skew symmetric matrix, then the value of $|a+b+c|$ is

Answer 1

Given: $A=\left[\begin{array}{ccc}0& a& 3\\ 2& b& -1\\ c& 1& 0\end{array}\right]$
Since, $A$ is skew-symmetric matrix.
$\therefore A^T=-A$
$\Rightarrow \left[\begin{array}{ccc}0& 2& c\\ a& b& 1\\ 3& -1& 0\end{array}\right]=-\left[\begin{array}{ccc}0& a& 3\\ 2& b& -1\\ c& 1& 0\end{array}\right]$
$\Rightarrow \left[\begin{array}{ccc}0& 2& c\\ a& b& 1\\ 3& -1& 0\end{array}\right]=\left[\begin{array}{ccc}0& -a& -3\\ -2& -b& 1\\ -c& -1& 0\end{array}\right]$
By equality of matrics , we get $a=-2,c=-3\text{and}b=-b$ $\Rightarrow b=0$
$\therefore a=-2,b=0$ and $c=-3$
$\Rightarrow |a+b+c|=|-2+0-3|=5$