Question

If A=$\left|\begin{array}{cc}1& 2\\ 4& 2\end{array}\right|$, then show that $|2A|=4|A|$

Answer 1

$A=\left[\begin{array}{cc}1& 2\\ 4& 2\end{array}\right]$
$\Rightarrow 2A=2\left[\begin{array}{cc}1& 2\\ 4& 2\end{array}\right]=\left[\begin{array}{cc}2& 4\\ 8& 4\end{array}\right]$
$\text{L.H.S}=|2A|=\left|\begin{array}{cc}2& 4\\ 8& 4\end{array}\right|=2\times 4-4\times 8=8-32=-24$
Now, $|A|=\left|\begin{array}{cc}1& 2\\ 4& 2\end{array}\right|=1\times 2-2\times 4=2-8=-6$
$\text{R.H.S}=4|A|=4\times (-6)=-24=\text{L.H.S.}$