If the matrix A=02K−1 satisfies AA3+3I=2I, then the value of K is

# If the matrix $\mathrm{A}=\left(\begin{array}{cc}0& 2\\ \mathrm{K}& -1\end{array}\right)$ satisfies $\mathrm{A}\left({\mathrm{A}}^{3}+3\mathrm{I}\right)=2\mathrm{I}$, then the value of K is

1. A

$\frac{1}{2}$

2. B

$-\frac{1}{2}$

3. C

$-1$

4. D

1

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### Solution:

Given matrix,

$\mathrm{A}=\left[\begin{array}{cc}0& 2\\ \mathrm{K}& -1\end{array}\right]$

Characteristic equation of A is

$\because$  Every square matrix satisfied its own characteristic equation.

Comparing Eqs. (i) and (ii), we get

$1+4\mathrm{K}=3⇒\mathrm{K}=\frac{1}{2}$

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