A real $3 \times 3$ matrix $A$ with elements $A_{i j}$ is said to be upper triangular if $A_{i j}=0$ whenever $i>j$. Prove that if $A$ and $B$ are upper triangular $3 \times 3$ real matrices then so is the matrix product $A B$.

Consider the matrix

$$A=\left(\begin{array}{rrr} 1 & 2 & 0 \\ 0 & -1 & 1 \\ 0 & 0 & -1 \end{array}\right)$$

Show that $A^{3}+A^{2}-A=I$. Write $A^{-1}$ as a linear combination of $A^{2}, A$ and $I$ and hence compute $A^{-1}$ explicitly.

For all integers $n$ (including negative integers), prove that there exist coefficients $\alpha_{n}, \beta_{n}$ and $\gamma_{n}$ such that

$$A^{n}=\alpha_{n} A^{2}+\beta_{n} A+\gamma_{n} I$$

For all integers $n$ (including negative integers), show that

$$\left(A^{n}\right)_{11}=1, \quad\left(A^{n}\right)_{22}=(-1)^{n}, \quad \text { and } \quad\left(A^{n}\right)_{23}=n(-1)^{n-1}$$

Hence derive a set of 3 simultaneous equations for $\left\{\alpha_{n}, \beta_{n}, \gamma_{n}\right\}$ and find their solution.