Question

If $M=\left[\begin{array}{cc}1& 2\\ 2& 3\end{array}\right]$ and $M^2-\lambda M-I=0$, then $\lambda$ equals

• A. $-2$
• B. $2$
• C. $-4$
• D. $4$

Answer 1

The correct option is D $4$

$M=\left[\begin{array}{cc}1& 2\\ 2& 3\end{array}\right]$ and $M^2-\lambda M-I_2=O$
$M^2-\lambda M-I_2=O$
$\Rightarrow \lambda M=M^2-I_2$
$\Rightarrow \lambda M=\left[\begin{array}{cc}5& 8\\ 8& 13\end{array}\right]-\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]$
$\Rightarrow \lambda M=\left[\begin{array}{cc}4& 8\\ 8& 12\end{array}\right]=4M$
$\therefore \lambda =4$
Hence, option D.