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, find $a,b$ and $c$

Answer 1

$\left[\begin{array}{ccc}0& a& 3\\ 2& b& -1\\ c& 1& 0\end{array}\right]$ is skew symmetric.

If a matrix is skew symmetric then

$A={-A}^T$

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

and $3=-c\Rightarrow c=-3$