If, then show that

Question:
If $A=\left[\begin{array}{ll}1 & 2 \\ 4 & 2\end{array}\right]$, then show that $|2 A|=4|A|$

Solution:

The given matrix is $A=\left[\begin{array}{ll}1 & 2 \\ 4 & 2\end{array}\right]$.

$\therefore 2 A=2\left[\begin{array}{ll}1 & 2 \\ 4 & 2\end{array}\right]=\left[\begin{array}{ll}2 & 4 \\ 8 & 4\end{array}\right]$

$\therefore$ L.H.S. $=|2 A|=\left|\begin{array}{ll}2 & 4 \\ 8 & 4\end{array}\right|=2 \times 4-4 \times 8=8-32=-24$

Now, $|A|=\left|\begin{array}{ll}1 & 2 \\ 4 & 2\end{array}\right|=1 \times 2-2 \times 4=2-8=-6$

$\therefore$ R.H.S $=4|A|=4 \times(-6)=-24$

$\therefore$ L.H.S. $=$ R.H.S.