Question

If \[A = \begin{bmatrix}2 & 3 \\ 5 & - 2\end{bmatrix}\]  be such that \[A^{- 1} = kA\], then k equals ___________ .

Options

• 19

• `1/19`

• -19

• `-1/19`

Answer 1

`1/19`

\[adj A = \begin{bmatrix}- 2 & - 3 \\ - 5 & 2\end{bmatrix}\]

\[\left| A \right| = - 19\]

\[ \therefore A^{- 1} = \frac{1}{\left| A \right|}adjA\]

\[ \Rightarrow A^{- 1} = - \frac{1}{19}\begin{bmatrix}- 2 & - 3 \\ - 5 & 2\end{bmatrix}\]

Now,

\[ A^{- 1} = kA\]

\[ \Rightarrow - \frac{1}{19}\begin{bmatrix}- 2 & - 3 \\ - 5 & 2\end{bmatrix} = kA\]

\[ \Rightarrow \frac{1}{19}\begin{bmatrix}2 & 3 \\ 5 & - 2\end{bmatrix} = kA\]

\[ \Rightarrow \frac{1}{19}A = kA\]

\[ \Rightarrow k = \frac{1}{19}\]