I'm trying to understand this pre-exam questions, but doesn't go well. Having the exam tomorrow. Can someone help me to understand what and how should I do here?
With help of $M=\begin{pmatrix} 3 & 2 \\ 1 & 4 \end{pmatrix}$ and $I=\begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}$ calculate the determinant of $\bigl(M-\lambda I \bigr)$
With help of $M=\begin{pmatrix} m_{11} & m_{12} \\ m_{21} & m_{22} \end{pmatrix}$ and $I=\begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}$ calculate the determinant of $\bigl(M-\lambda I \bigr)$
Prove Cayley-Hamilton's rate for arbitrary 2 × 2 matrices: $M=\begin{pmatrix} m_{11} & m_{12} \\ m_{21} & m_{22} \end{pmatrix}$ by direct calculation.
For $M=\begin{pmatrix} m_{11} & m_{12} \\ m_{21} & m_{22} \end{pmatrix}$ we have
$\det(M- \lambda I)=\lambda^2-(m_{11}+m_{22}) \lambda -m_{12}m_{21}$.
You have to show that
$$M^2-(m_{11}+m_{22})M -m_{12}m_{21}I=0.$$