Question

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

Answer 1

Given,

$M=\left[\begin{array}{cc}1& 2\\ 2& 3\end{array}\right]\therefore M^2=\left[\begin{array}{cc}1& 2\\ 2& 3\end{array}\right]\times \left[\begin{array}{cc}1& 2\\ 2& 3\end{array}\right]=\left[\begin{array}{cc}5& 8\\ 8& 13\end{array}\right]$

Consider,

$M^2-\lambda M-I=O$

$\Rightarrow \left[\begin{array}{cc}5-\lambda -1& 8-2\lambda \\ 8-2\lambda & 13-3\lambda -1\end{array}\right]=\left[\begin{array}{cc}0& 0\\ 0& 0\end{array}\right]$

$\Rightarrow 8-2\lambda =0$

$\Rightarrow \lambda =4$