This problem involves matrix multiplication and solving a system of equations derived from a matrix equality. The core idea is to compute powers of the given matrix $A$ and then equate the resulting matrix $A^3$ with $A$ itself, element by element. This will lead to a system of algebraic equations involving the unknown parameters $a$ and $c$, which we then solve.

1. Key Concepts and Formulas

• Matrix Multiplication: For two matrices $P$ (of dimension $m \times k$) and $Q$ (of dimension $k \times n$), their product $PQ$ is an $m \times n$ matrix where the element in the $i$-th row and $j$-th column, $(PQ)_{ij}$, is given by the dot