Question

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

• A. $ 19 $
• B. $ \frac{1}{19} $
• C. $ -19 $
• D. $ - \frac {1}{19} $

Answer 1

Answer: (B)

Given, $A=\left[\begin{matrix}2&3\\ 5&-2\end{matrix}\right]$
$\therefore A^{-1}=\frac{1}{-4-15}\left[\begin{matrix}-2&-3\\ -5&2\end{matrix}\right]$
$=\frac{-1}{19}\left[\begin{matrix}-2&-3\\ -5&2\end{matrix}\right]=\frac{1}{19}\left[\begin{matrix}2&3\\ 5&-2\end{matrix}\right]$
[multiplying $-1$ each element of a matrix]
$=\frac{1}{19}A$
Given, $A^{-1}=kA$
$\therefore k=\frac{1}{19}$