1. ## Help with matrices

Let A denote the 2x2 matrix with $a_{11} = 2, a_{12} = 0, a_{21} = 4$ and $a_{22} = 1$. Find a polynomial p(x) with degree two such that p(A) = 0

How can I find this one?

And how can I prove that $(A+B)^2 = A^2 + 2AB + B^2$ and $(A-B)(A+B) = A^2 - B^2$, if A and B are square matrices of the same size.

Thank you.

2. Use the Cayley-Hamilton identity:

$A^2-Tr(A)\cdot A+\det(A)\cdot I_2=O_2$

where $Tr(A)=a_{11}+a_{22}, \ \det(A)=a_{11}a_{22}-a_{12}a_{21}$

and $I_2=\begin{pmatrix}1 & 0\\0 & 1\end{pmatrix}, \ O_2=\begin{pmatrix}0 & 0\\0 & 0\end{pmatrix}$