Question

With out expanding show that $\left|\begin{array}{ccc}0& 2& -3\\ -2& 0& 4\\ 3& -4& 0\end{array}\right|=0$.

Answer 1

Let $A=$$\left|\begin{array}{ccc}0& 2& -3\\ -2& 0& 4\\ 3& -4& 0\end{array}\right|$

Now $A^T=$ $\left|\begin{array}{ccc}0& -2& 3\\ 2& 0& -4\\ -3& 4& 0\end{array}\right|$

or, $A^T=$ $(-1{)}^3\left|\begin{array}{ccc}0& 2& -3\\ -2& 0& 4\\ 3& -4& 0\end{array}\right|$ [ Taking $-1$ common from the first, second and third row respectively]

or, $A=-A$ [ Since $A^T=A$ as $A$ is the determinant]

or, $2A=0$

or, $A=0$.