If A= 1&2 4&2 , then show that |2A |=4 |A | - Vidyadip

Question

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

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Answer

The given matrix is $\text{A}=\begin{bmatrix}1&2\\4&2\end{bmatrix}$
$\therefore2\text{A}=2\begin{bmatrix}1&2\\4&2\end{bmatrix}=\begin{bmatrix}2&4\\8&4\end{bmatrix}$
$\therefore\text{L.H.S.}=|2\text{A}|=\begin{vmatrix}2&4\\8&4\end{vmatrix}=2\times4-4\times8=8-32=-24$
$\text{Now},|\text{A}|=\begin{vmatrix}1&2\\4&2\end{vmatrix}=2-8=-6$
$\therefore\text{R.H.S.}=4|\text{A}|=4\times\left(-6\right)=-24$
$\therefore\text{L.H.S.}=\text{R.H.S.}$

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