## A Recursive Relationship for a Power of a Matrix

## Problem 685

Suppose that the $2 \times 2$ matrix $A$ has eigenvalues $4$ and $-2$. For each integer $n \geq 1$, there are real numbers $b_n , c_n$ which satisfy the relation

\[ A^{n} = b_n A + c_n I , \]
where $I$ is the identity matrix.

Find $b_n$ and $c_n$ for $2 \leq n \leq 5$, and then find a recursive relationship to find $b_n, c_n$ for every $n \geq 1$.

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