If $A=\begin{bmatrix}2 & 5\\3 & -2\end{bmatrix}$ be such that $A^{-1}=kA,$ then k is equal to :

$\frac{1}{19}$

The correct answer is option (4) → $\frac{1}{19}$

$A^{-1}=kA$

so $A^{-1}×A=kA×A$

$I=kA^2$

$A^2=\begin{bmatrix}2 & 5\\3 & -2\end{bmatrix}\begin{bmatrix}2 & 5\\3 & -2\end{bmatrix}=\begin{bmatrix}19 & 0\\0 & 19\end{bmatrix}=19\begin{bmatrix}1 & 0\\0 & 1\end{bmatrix}$

so $k=\frac{1}{19}$