Cubing a Matrix Without Cubing Its Entries

$A = \begin{bmatrix} 1 & 2 \\ 3 & -1 \end{bmatrix}$

Here $A^3$ means the matrix product $A \cdot A \cdot A$ — three copies of $A$ multiplied together, using matrix multiplication each time.

Compute $A^3$.

Suggestion: work out $A^2$ completely and look hard at what you get before charging into the third multiplication.