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 $A^k$.
Add to solve later Let $A$ be an $n \times n$ matrix. Suppose that all the eigenvalues $\lambda$ of $A$ are real and satisfy $\lambda <1$.
Then show that the determinant \[ \det(I-A) >0,\] where $I$ is the $n \times n$ identity matrix.
Add to solve later 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 $B^2=A$ where
\[A=\begin{bmatrix}
1 & -1 & 0 \\
-1 &2 &-1 \\
0 & -1 & 1
\end{bmatrix}\,\,\,\,?\]
(Princeton University Linear Algebra Exam)
Add to solve later 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 $A=I_n$.
Add to solve later 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.
Add to solve later An $n\times n$ matrix $A$ is called nilpotent if $A^k=O$, where $O$ is the $n\times n$ zero matrix.
Prove the followings.
(a) The matrix $A$ is nilpotent if and only if all the eigenvalues of $A$ is zero.
(b) The matrix $A$ is nilpotent if and only if $A^n=O$.
Read solution
Add to solve later Let $A$ be an $n\times n$ matrix and let $\lambda_1, \dots, \lambda_n$ be its eigenvalues.
Show that
(1) $$\det(A)=\prod_{i=1}^n \lambda_i$$
(2) $$\tr(A)=\sum_{i=1}^n \lambda_i$$
Here $\det(A)$ is the determinant of the matrix $A$ and $\tr(A)$ is the trace of the matrix $A$.
Namely, prove that (1) the determinant of $A$ is the product of its eigenvalues, and (2) the trace of $A$ is the sum of the eigenvalues.
Read solution
Add to solve later 
