Question

If $A=\left[\begin{array}{cc}1& 2\\ 2& 3\end{array}\right]$ and $A^2-KA-I_2=0,$ then value of k is-

• A. $4$
• B. $2$
• C. $1$
• D. $-4$

Answer 1

Answer: (A)

$4$

$A=\left[\begin{array}{cc}1& 2\\ 2& 3\end{array}\right]$

characteristic $e{q}^n$ is given by $|A-\lambda I|=0$, where

I is identity matrix of order $2\times 2$.

$\therefore A-\lambda I=\left[\begin{array}{cc}1-\lambda & 2\\ 2& 3-\lambda \end{array}\right]$

$\therefore |A-\lambda I|=(1-\lambda )(3-\lambda )-4$

$\therefore {\lambda }^2-4\lambda -1=0$

Replacing $\lambda$ by matrix A

$A^2-4A-1=0$

$A^2-4A-1\times I=0$

on comparing

k=4