We fist show that the eigenvalues of $A$ are $r$-th roots of unity.
Let $\lambda$ be an eigenvalue of $A$ and let $\mathbf{x}$ be an eigenvector corresponding to $\lambda$. That is, $A\mathbf{x}=\lambda \mathbf{x}$.

Multiplying this equality by $A$ on the left, we see that
\[A^2 \mathbf{x}=\lambda A\mathbf{x}=\lambda^2 \mathbf{x}.\]
Again multiplying this equality by $A$ on the left we get $A^3\mathbf{x}=\lambda^3\mathbf{x}$.

Inductively, we obtain $A^r \mathbf{x}=\lambda^r \mathbf{x}$.
Since $A^r=I_n$, we get $\mathbf{x}=\lambda^r \mathbf{x}$, Thus $(\lambda^r-1)\mathbf{x}=\mathbf{0}$ and since $\mathbf{x}$ is nonzero vector (as it is an eigenvector), we have $\lambda^r=1$. Therefore eigenvalues are $r$-th roots of unity.

Next, we consider the Jordan canonical form of $A$. There exists an invertible matrix $P$ such that $P^{-1}AP$ is an upper triangular matrix $T=(t_{ij})$ whose diagonal entries are eigenvalues of $A$.
Since the trace of $A$ is equal to the trace of $P^{-1}AP$, we see that $\tr(A)$ is a sum of $n$ $r$-th roots of unity. Thus
\begin{align*}
|\tr(A)|=|\sum_{i=1}^{n}t_{ii}| \leq \sum_{i=1}^{n}|t_{ii}|=n. \tag{*}
\end{align*}

(b) If $|\tr(A)|=n$, then $A=\zeta I_n$ for an $r$-th root of unity $\zeta$

We first refine the proof of part (a).
Note that the matrix $A$ is a solution of the equation $x^r-1$,

So the minimal polynomial of $A$ divides $x^r-1$, thus the minimal polynomial has no repeated roots. Thereby the Jordan canonical form $T$ is actually a diagonal matrix.

Now we start the proof of (b). The inequality in (*) becomes the equality if and only if all the roots of unity $t_{ii}$ are the same root of unity $\zeta$.
Then the Jordan canonical form is $T=\zeta I_n$, hence $A=\zeta I_n$.

(c) $\tr(A)=n$ if and only if $A=I_n$

By part (b), we have $A=\zeta I_n$ for some $r$-th root of unity $\zeta$ but then $n=\tr(A)=\zeta n$ implies $\zeta=1$. Thus $A=I_n$ as required.

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 […]

If Every Trace of a Power of a Matrix is Zero, then the Matrix is Nilpotent
Let $A$ be an $n \times n$ matrix such that $\tr(A^n)=0$ for all $n \in \N$.
Then prove that $A$ is a nilpotent matrix. Namely there exist a positive integer $m$ such that $A^m$ is the zero matrix.
Steps.
Use the Jordan canonical form of the matrix $A$.
We want […]

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 […]

If a Power of a Matrix is the Identity, then the Matrix is Diagonalizable
Let $A$ be an $n \times n$ complex matrix such that $A^k=I$, where $I$ is the $n \times n$ identity matrix.
Show that the matrix $A$ is diagonalizable.
Hint.
Use the fact that if the minimal polynomial for the matrix $A$ has distinct roots, then $A$ is […]

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. […]

A Square Root Matrix of a Symmetric Matrix
Answer the following two questions with justification.
(a) Does there exist a $2 \times 2$ matrix $A$ with $A^3=O$ but $A^2 \neq O$? Here $O$ denotes the $2 \times 2$ zero matrix.
(b) Does there exist a $3 \times 3$ real matrix $B$ such that $B^2=A$ […]

Find All the Eigenvalues of $A^k$ from Eigenvalues of $A$
Let $A$ be $n\times n$ matrix and let $\lambda_1, \lambda_2, \dots, \lambda_n$ be all the eigenvalues of $A$. (Some of them may be the same.)
For each positive integer $k$, prove that $\lambda_1^k, \lambda_2^k, \dots, \lambda_n^k$ are all the eigenvalues of […]

Solve the following system of linear equations using Gauss-Jordan elimination. \begin{align*} 6x+8y+6z+3w &=-3 \\ 6x-8y+6z-3w &=3\\ 8y \,\,\,\,\,\,\,\,\,\,\,- 6w &=6...

## 2 Responses

[…] a similar problem, also see Finite order matrix and its trace. This problem is kind of the converse of the current […]

[…] also Finite order matrix and its trace for a similar […]