Suppose that $A$ is a diagonalizable $n\times n$ matrix and has only $1$ and $-1$ as eigenvalues.
Show that $A^2=I_n$, where $I_n$ is the $n\times n$ identity matrix.

Diagonalize the matrix $A$ so that the diagonal matrix has $\pm 1$ on diagonal entries.

Proof.

Since $A$ is diagonalizable, there exists an invertible matrix $P$ such that $P^{-1}AP=D$, where $D$ is a diagonal matrix.
Since $A$ has only $\pm 1$ as eigenvalues, we can choose $P$ so that the diagonal entries of $D$ are either $\pm 1$.

Then we have $A=PDP^{-1}$ and \(\require{cancel}\)
\begin{align*}
A^2=(PDP^{-1})(PDP^{-1})=(PD\cancel{P}^{-1})(\cancel{P}DP^{-1})=PD^2P^{-1}.
\end{align*}
Note that $D^{2}=I_2$ and thus we have $A^2=I_2$ as required.

Generalization

As a generalization, consider the following problem.

Problem.
Suppose that $A$ is a diagonalizable $n\times n$ matrix and the eigenvalues of $A$ are $r$-th roots of unity for some positive integer $r$.
Show that $A^r=I_n$, where $I_n$ is the $n\times n$ identity matrix.

The proof of Problem 2 is a straightforward generalization of the proof of Problem 1. So give it a shot.

Two Matrices with the Same Characteristic Polynomial. Diagonalize if Possible.
Let
\[A=\begin{bmatrix}
1 & 3 & 3 \\
-3 &-5 &-3 \\
3 & 3 & 1
\end{bmatrix} \text{ and } B=\begin{bmatrix}
2 & 4 & 3 \\
-4 &-6 &-3 \\
3 & 3 & 1
\end{bmatrix}.\]
For this problem, you may use the fact that both matrices have the same characteristic […]

Given Eigenvectors and Eigenvalues, Compute a Matrix Product (Stanford University Exam)
Suppose that $\begin{bmatrix}
1 \\
1
\end{bmatrix}$ is an eigenvector of a matrix $A$ corresponding to the eigenvalue $3$ and that $\begin{bmatrix}
2 \\
1
\end{bmatrix}$ is an eigenvector of $A$ corresponding to the eigenvalue $-2$.
Compute $A^2\begin{bmatrix}
4 […]

True of False Problems on Determinants and Invertible Matrices
Determine whether each of the following statements is True or False.
(a) If $A$ and $B$ are $n \times n$ matrices, and $P$ is an invertible $n \times n$ matrix such that $A=PBP^{-1}$, then $\det(A)=\det(B)$.
(b) If the characteristic polynomial of an $n \times n$ matrix $A$ […]

Is an Eigenvector of a Matrix an Eigenvector of its Inverse?
Suppose that $A$ is an $n \times n$ matrix with eigenvalue $\lambda$ and corresponding eigenvector $\mathbf{v}$.
(a) If $A$ is invertible, is $\mathbf{v}$ an eigenvector of $A^{-1}$? If so, what is the corresponding eigenvalue? If not, explain why not.
(b) Is $3\mathbf{v}$ an […]

If the Kernel of a Matrix $A$ is Trivial, then $A^T A$ is Invertible
Let $A$ be an $m \times n$ real matrix.
Then the kernel of $A$ is defined as $\ker(A)=\{ x\in \R^n \mid Ax=0 \}$.
The kernel is also called the null space of $A$.
Suppose that $A$ is an $m \times n$ real matrix such that $\ker(A)=0$. Prove that $A^{\trans}A$ is […]

A Square Root Matrix of a Symmetric Matrix with Non-Negative Eigenvalues
Let $A$ be an $n\times n$ real symmetric matrix whose eigenvalues are all non-negative real numbers.
Show that there is an $n \times n$ real matrix $B$ such that $B^2=A$.
Hint.
Use the fact that a real symmetric matrix is diagonalizable by a real orthogonal matrix.
[…]

Finite Order Matrix and its Trace
Let $A$ be an $n\times n$ matrix and suppose that $A^r=I_n$ for some positive integer $r$. Then show that
(a) $|\tr(A)|\leq n$.
(b) If $|\tr(A)|=n$, then $A=\zeta I_n$ for an $r$-th root of unity $\zeta$.
(c) $\tr(A)=n$ if and only if $A=I_n$.
Proof.
(a) […]