Use the fact that if the minimal polynomial for the matrix $A$ has distinct roots, then $A$ is diagonalizable.

Proof.

Since the matrix $A$ satisfies the equation $x^k-1$, the minimal polynomial of $A$ divides $x^k-1$.
Since
\[x^k-1=\prod_{j=0}^{k-1}(x-e^{2\pi i j/k}),\]
the roots of $x^k-1$ are all distinct.

Hence the roots of the minimal polynomial are also distinct.
Therefore the matrix $A$ is diagonalizable.

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) […]

Extension Degree of Maximal Real Subfield of Cyclotomic Field
Let $n$ be an integer greater than $2$ and let $\zeta=e^{2\pi i/n}$ be a primitive $n$-th root of unity. Determine the degree of the extension of $\Q(\zeta)$ over $\Q(\zeta+\zeta^{-1})$.
The subfield $\Q(\zeta+\zeta^{-1})$ is called maximal real subfield.
Proof. […]

Trace of the Inverse Matrix of a Finite Order Matrix
Let $A$ be an $n\times n$ matrix such that $A^k=I_n$, where $k\in \N$ and $I_n$ is the $n \times n$ identity matrix.
Show that the trace of $(A^{-1})^{\trans}$ is the conjugate of the trace of $A$. That is, show that […]

Diagonalizable Matrix with Eigenvalue 1, -1
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.
(Stanford University Linear Algebra Exam)
See below for a generalized problem.
Hint.
Diagonalize the […]

How to Find Eigenvalues of a Specific Matrix.
Find all eigenvalues of the following $n \times n$ matrix.
\[
A=\begin{bmatrix}
0 & 0 & \cdots & 0 &1 \\
1 & 0 & \cdots & 0 & 0\\
0 & 1 & \cdots & 0 &0\\
\vdots & \vdots & \ddots & \ddots & \vdots \\
0 & […]

A Matrix Similar to a Diagonalizable Matrix is Also Diagonalizable
Let $A, B$ be matrices. Show that if $A$ is diagonalizable and if $B$ is similar to $A$, then $B$ is diagonalizable.
Definitions/Hint.
Recall the relevant definitions.
Two matrices $A$ and $B$ are similar if there exists a nonsingular (invertible) matrix $S$ such […]

Degree of an Irreducible Factor of a Composition of Polynomials
Let $f(x)$ be an irreducible polynomial of degree $n$ over a field $F$. Let $g(x)$ be any polynomial in $F[x]$.
Show that the degree of each irreducible factor of the composite polynomial $f(g(x))$ is divisible by $n$.
Hint.
Use the following fact.
Let $h(x)$ is an […]

Diagonalizable by an Orthogonal Matrix Implies a Symmetric Matrix
Let $A$ be an $n\times n$ matrix with real number entries.
Show that if $A$ is diagonalizable by an orthogonal matrix, then $A$ is a symmetric matrix.
Proof.
Suppose that the matrix $A$ is diagonalizable by an orthogonal matrix $Q$.
The orthogonality of the […]

[…] Remark that if $A$ is a square matrix over $C$ with $A^k=I$, then $A$ is diagonalizable. For a proof of this fact, see If a power of a matrix is the identity, then the matrix is diagonalizable […]

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[…] Remark that if $A$ is a square matrix over $C$ with $A^k=I$, then $A$ is diagonalizable. For a proof of this fact, see If a power of a matrix is the identity, then the matrix is diagonalizable […]