If $X$ is a symmetric matrix with $X^s = X^{s+1}$ where $s$ is an integer greater than or equal to $1$. Show that $X$ is idempotent.
Attempt:
$X^s = X^{s+1}$
⇒ $XX^s = XX^{s+1}$
⇒ $X^{s+1}$ = $X^{s+2}$
⇒ $XX^s = X^sX^2$
⇒ $X = (X^s)^{-1}X^sX^2$
⇒ $X = IX^2$
⇒ $X = X^2$, therefore $X$ is idempotent.
What makes me think this is wrong is that I haven't used the fact that $X$ is symmetric, which makes me wonder why it is stated in the question. Also it seemed a little too simple and it's easy to make mistakes with Matrix algebra. Would greatly appreciate it if someone can confirm that this is the way to answer this question.
You assumed that $X$ is invertible. This is not allowed.
The proof goes like this:
$X$ is symmetric, hence $X$ is diagonalizable, hence the minimal polnomial splits into distinct linear factors. On the other hand the minimal polynomial is a divisor of $t^s(t-1)$. Thus it is a divisor of $t(t-1)=t^2-t$. Hence $X$ is idempotent.
Actually the proof shows: If $X$ is diagonalizable and $X^s=X^{s+1}$ for some $s \geq 1$, then $X$ is idempotent.