Question

If the matrix $A=\left[\begin{array}{ccc}0& -2& 1\\ b& a& 4\\ c& -4& 0\end{array}\right]$ is skew-symmetric, then the values of $a+b+c$ is

$\left(a\right)1\left(b\right)-1$
$\left(c\right)0\left(d\right)2$

Answer 1

$A=\left[\begin{array}{ccc}0& -2& 1\\ b& a& 4\\ c& -4& 0\end{array}\right]$ is skew-symmetric.

$\therefore A^T=-A$

$\Rightarrow {\left[\begin{array}{ccc}0& -2& 1\\ b& a& 4\\ c& -4& 0\end{array}\right]}^T=-\left[\begin{array}{ccc}0& -2& 1\\ b& a& 4\\ c& -4& 0\end{array}\right]$
$\Rightarrow \left[\begin{array}{ccc}0& b& c\\ -2& a& -4\\ 1& 4& 0\end{array}\right]=\left[\begin{array}{ccc}0& 2& -1\\ -b& -a& -4\\ -c& 4& 0\end{array}\right]$

$\Rightarrow a=0,b=2,c=-1$
$\Rightarrow a+b+c=1$
Ans : $\left(a\right)$