Let A= 2 & 3 5 & -2 be such that A^ -1 =k A, then k equals: - Vidyadip

MCQ

Let $\text{A}=\begin{bmatrix} 2 & 3 \\ 5 & -2 \end{bmatrix}$ be such that $A^{-1}=k A,$ then $k$ equals:

A
$19$
✓
$\frac{1}{19}$
C
$-19$
D
$-\frac{1}{19}$

✓

Answer

Correct option: B.

$\frac{1}{19}$

$\text{adj A}=\begin{bmatrix} -2 & -3 \\ -5 & 2 \end{bmatrix}$
$|\text{A}|=-19$
$\therefore\ \text{A}^{-1}=-\frac{1}{|\text{A}|}\text{ adj A}$
$\Rightarrow\ \text{A}^{-1}=-\frac{1}{19}\begin{bmatrix} -2 & -3 \\ -5 & 2 \end{bmatrix}$
Now,
$A^{-1}=k A$
$\Rightarrow-\frac{1}{19}\begin{bmatrix} -2 & -3 \\ -5 & 2 \end{bmatrix}=\text{kA}$
$\Rightarrow\frac{1}{19}\begin{bmatrix} 2 & 3 \\ 5 & -2 \end{bmatrix}=\text{kA}$
$\Rightarrow\frac{1}{19}\text{A}=\text{kA}$
$\Rightarrow\text{k}=\frac{1}{19}$

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