Question

If ${\Delta }_1=\left|\begin{array}{ccc}1& 0& 2\\ 0& 2& -1\\ 0& -1& 3\end{array}\right|$ and ${\Delta }_2=\left|\begin{array}{ccc}2& -1& 0\\ 3& 1& 0\\ 0& 0& 2\end{array}\right|,$ then the value of ${\Delta }_1{\Delta }_2$ is

• A. $100$
• B. $-100$
• C. $50$
• D. $-50$

Answer 1

The correct option is C $50$

Given${\Delta }_1=\left|\begin{array}{ccc}1& 0& 2\\ 0& 2& -1\\ 0& -1& 3\end{array}\right|,{\Delta }_2=\left|\begin{array}{ccc}2& -1& 0\\ 3& 1& 0\\ 0& 0& 2\end{array}\right|$
$\Rightarrow {\Delta }_1{\Delta }_2=\left|\begin{array}{ccc}2+0+0& -1+0+0& 0+0+4\\ 0+6+0& 0+2+0& 0+0-2\\ 0-3+0& 0-1+0& 0-0+6\end{array}\right|$
So, ${\Delta }_1{\Delta }_2=\left|\begin{array}{ccc}2& -1& 4\\ 6& 2& -2\\ -3& -1& 6\end{array}\right|$
Expanding along $R_1,$

$\Rightarrow {\Delta }_1{\Delta }_2=2(12-2)+1(36-6)+4(-6+6)=50$