Question
If $A=\left[\begin{array}{cc}2 & 3 \\ 5 & -2\end{array}\right]$, be such that $A^{-1}=k A$, then find the value of $k$.
✓
Answer
$\text{Finding}{~} A^{-1}=\frac{-1}{19}\left[\begin{array}{cc}-2 & -3 \\ -5 & 2\end{array}\right]$
$\Rightarrow \frac{-1}{19}\left[\begin{array}{cc}-2 & -3 \\ -5 & 2\end{array}\right]=\left[\begin{array}{cc}2 k & 3 k \\ 5 k & -2 k\end{array}\right]$
$\Rightarrow k=\frac {1}{19}$
$\Rightarrow \frac{-1}{19}\left[\begin{array}{cc}-2 & -3 \\ -5 & 2\end{array}\right]=\left[\begin{array}{cc}2 k & 3 k \\ 5 k & -2 k\end{array}\right]$
$\Rightarrow k=\frac {1}{19}$
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