Question

If the matrix $A=\left[\begin{array}{ccc}0& a& -3\\ 2& 0& -1\\ b& 1& 0\end{array}\right]$ is skew symmetric, find the values of $a$ and $b$.

Answer 1

Here, $A$ is a skew symmetric matrix
$\therefore A^{\prime }=-A$

${\left[\begin{array}{ccc}0& a& -3\\ 2& 0& -1\\ b& 1& 0\end{array}\right]}^{\prime }=-\left[\begin{array}{ccc}0& a& -3\\ 2& 0& -1\\ b& 1& 0\end{array}\right]$

$\Rightarrow \left[\begin{array}{ccc}0& 2& b\\ a& 0& 1\\ -3& -1& 0\end{array}\right]=\left[\begin{array}{ccc}0& -a& 3\\ -2& 0& 1\\ -b& -1& 0\end{array}\right]$

Comparing both sides, we get,

$\Rightarrow a=-2$ and $b=3$

Alternate solution:
$A=\left[a_{ij}\right]$ $\text{is a skew symmetric matrix then}$
$a_{ij}=-a_{ji}$
$\text{therefore using this we know that}$,
$a=-2\text{and}b=3$