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If $M=\left[\begin{array}{ll}1 & 2 \\ 2 & 3\end{array}\right]$ and $M^2-\lambda M-I_2=0$, then $\lambda=$
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$4$
$M^2-\lambda M-I_2=0$
$\Rightarrow\left[\begin{array}{ll}1 & 2 \\ 2 & 3\end{array}\right]\left[\begin{array}{ll}1 & 2 \\ 2 & 3\end{array}\right]-\left[\begin{array}{cc}\lambda & 2 \lambda \\ 2 \lambda & 3 \lambda\end{array}\right]-\left[\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right]=0$
$\Rightarrow\left[\begin{array}{cc}5 & 8 \\ 8 & 13\end{array}\right]-\left[\begin{array}{cc}\lambda & 2 \lambda \\ 2 \lambda & 3 \lambda\end{array}\right]-\left[\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right]=0$
$\Rightarrow\left[\begin{array}{cc}5-\lambda & 8-2 \lambda \\ 8-2 \lambda & 13-3 \lambda\end{array}\right]=\left[\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right]$
$\Rightarrow 5-\lambda=1,8-2 \lambda=0,13-3 \lambda=1$
$\Rightarrow \lambda=4$, which satisfies|all the three equations.
$\Rightarrow\left[\begin{array}{ll}1 & 2 \\ 2 & 3\end{array}\right]\left[\begin{array}{ll}1 & 2 \\ 2 & 3\end{array}\right]-\left[\begin{array}{cc}\lambda & 2 \lambda \\ 2 \lambda & 3 \lambda\end{array}\right]-\left[\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right]=0$
$\Rightarrow\left[\begin{array}{cc}5 & 8 \\ 8 & 13\end{array}\right]-\left[\begin{array}{cc}\lambda & 2 \lambda \\ 2 \lambda & 3 \lambda\end{array}\right]-\left[\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right]=0$
$\Rightarrow\left[\begin{array}{cc}5-\lambda & 8-2 \lambda \\ 8-2 \lambda & 13-3 \lambda\end{array}\right]=\left[\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right]$
$\Rightarrow 5-\lambda=1,8-2 \lambda=0,13-3 \lambda=1$
$\Rightarrow \lambda=4$, which satisfies|all the three equations.
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