Science:Math Exam Resources/Courses/MATH152/April 2017/Question B 06 (c)/Solution 1

Geometrically, when you reflects a vector $n$-times with respect to some plane, it becomes itself for even $n$ and become its reflection for odd $n$.

Since the matrix $A$ is the operation reflecting a vector in some plane (obtained in part (b)), for any vector $v$ in $\mathbb {R} ^{3}$, $A^{n}v=v$ for even $n$ and $A^{n}v=Av$ for odd $n$.

Therefore, we have

$\color {blue}{\begin{aligned}A^{n}={\begin{cases}A&n{\text{ odd}}\\I&n{\text{ even}}\end{cases}}\end{aligned}}$