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