(a) Let $A$ be a real orthogonal $n\times n$ matrix. Prove that the length (magnitude) of each eigenvalue of $A$ is $1$.
(b) Let $A$ be a real orthogonal $3\times 3$ matrix and suppose that the determinant of $A$ is $1$. Then prove that $A$ has $1$ as an eigenvalue.
Add to solve later Let $n$ be an odd integer and let $A$ be an $n\times n$ real matrix.
Prove that the matrix $A$ has at least one real eigenvalue.
Add to solve later Let $A$ be an $n\times n$ real matrix.
Prove that if $\lambda$ is an eigenvalue of $A$, then its complex conjugate $\bar{\lambda}$ is also an eigenvalue of $A$.
Add to solve later (a) Is the matrix $A=\begin{bmatrix}
1 & 2\\
0& 3
\end{bmatrix}$ similar to the matrix $B=\begin{bmatrix}
3 & 0\\
1& 2
\end{bmatrix}$?
(b) Is the matrix $A=\begin{bmatrix}
0 & 1\\
5& 3
\end{bmatrix}$ similar to the matrix $B=\begin{bmatrix}
1 & 2\\
4& 3
\end{bmatrix}$?
(c) Is the matrix $A=\begin{bmatrix}
-1 & 6\\
-2& 6
\end{bmatrix}$ similar to the matrix $B=\begin{bmatrix}
3 & 0\\
0& 2
\end{bmatrix}$?
(d) Is the matrix $A=\begin{bmatrix}
-1 & 6\\
-2& 6
\end{bmatrix}$ similar to the matrix $B=\begin{bmatrix}
1 & 2\\
-1& 4
\end{bmatrix}$?
Add to solve later Prove that if $A$ and $B$ are similar matrices, then their determinants are the same.
Add to solve later 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. Its only eigenvalues are $1, 2, 3, 4, 5$, possibly with multiplicities.
What is the nullity of the matrix $A+I_n$, where $I_n$ is the $n\times n$ identity matrix?
(The Ohio State University, Linear Algebra Final Exam Problem)
Add to solve later Let $A$ be an $n\times n$ matrix with the characteristic polynomial
\[p(t)=t^3(t-1)^2(t-2)^5(t+2)^4.\]
Assume that the matrix $A$ is diagonalizable.
(a) Find the size of the matrix $A$.
(b) Find the dimension of the eigenspace $E_2$ corresponding to the eigenvalue $\lambda=2$.
(c) Find the nullity of $A$.
(The Ohio State University, Linear Algebra Final Exam Problem)
Read solution
Add to solve later Consider the matrix
\[A=\begin{bmatrix}
3/2 & 2\\
-1& -3/2
\end{bmatrix} \in M_{2\times 2}(\R).\]
(a) Find the eigenvalues and corresponding eigenvectors of $A$.
(b) Show that for $\mathbf{v}=\begin{bmatrix}
1 \\
0
\end{bmatrix}\in \R^2$, we can choose $n$ large enough so that the length $\|A^n\mathbf{v}\|$ is as small as we like.
(University of California, Berkeley, Linear Algebra Final Exam Problem)
Read solution
Add to solve later Find all the eigenvalues and eigenvectors of the matrix
\[A=\begin{bmatrix}
3 & 9 & 9 & 9 \\
9 &3 & 9 & 9 \\
9 & 9 & 3 & 9 \\
9 & 9 & 9 & 3
\end{bmatrix}.\]
(Harvard University, Linear Algebra Final Exam Problem)
Add to solve later Let $A$ be an $n \times n$ matrix and let $c$ be a complex number.
(a) For each eigenvalue $\lambda$ of $A$, prove that $\lambda+c$ is an eigenvalue of the matrix $A+cI$, where $I$ is the identity matrix. What can you say about the eigenvectors corresponding to $\lambda+c$?
(b) Prove that the algebraic multiplicity of the eigenvalue $\lambda$ of $A$ is the same as the algebraic multiplicity of the eigenvalue $\lambda+c$ of $A+cI$ are equal.
(c) How about geometric multiplicities?
Add to solve later (a) Let
\[A=\begin{bmatrix}
0 & 0 & 0 & 0 \\
1 &1 & 1 & 1 \\
0 & 0 & 0 & 0 \\
1 & 1 & 1 & 1
\end{bmatrix}.\]
Find the eigenvalues of the matrix $A$. Also give the algebraic multiplicity of each eigenvalue.
(b) Let
\[A=\begin{bmatrix}
0 & 0 & 0 & 0 \\
1 &1 & 1 & 1 \\
0 & 0 & 0 & 0 \\
1 & 1 & 1 & 1
\end{bmatrix}.\]
One of the eigenvalues of the matrix $A$ is $\lambda=0$. Find the geometric multiplicity of the eigenvalue $\lambda=0$.
Add to solve later Let
\[A=\begin{bmatrix}
3 & -12 & 4 \\
-1 &0 &-2 \\
-1 & 5 & -1
\end{bmatrix}.\]
Then find all eigenvalues of $A^5$. If $A$ is invertible, then find all the eigenvalues of $A^{-1}$.
Add to solve later Let $A$ be an $n\times n$ complex matrix.
Let $p(x)=\det(xI-A)$ be the characteristic polynomial of $A$ and write it as
\[p(x)=x^n+a_{n-1}x^{n-1}+\cdots+a_1x+a_0,\]
where $a_i$ are real numbers.
Let $C$ be the companion matrix of the polynomial $p(x)$ given by
\[C=\begin{bmatrix}
0 & 0 & \dots & 0 &-a_0 \\
1 & 0 & \dots & 0 & -a_1 \\
0 & 1 & \dots & 0 & -a_2 \\
\vdots & & \ddots & & \vdots \\
0 & 0 & \dots & 1 & -a_{n-1}
\end{bmatrix}=
[\mathbf{e}_2, \mathbf{e}_3, \dots, \mathbf{e}_n, -\mathbf{a}],\]
where $\mathbf{e}_i$ is the unit vector in $\C^n$ whose $i$-th entry is $1$ and zero elsewhere, and the vector $\mathbf{a}$ is defined by
\[\mathbf{a}=\begin{bmatrix}
a_0 \\
a_1 \\
\vdots \\
a_{n-1}
\end{bmatrix}.\]
Then prove that the following two statements are equivalent.
Add to solve later Let $A, B$ be complex $2\times 2$ matrices satisfying the relation
\[A=AB-BA.\]
Prove that $A^2=O$, where $O$ is the $2\times 2$ zero matrix.
Add to solve later Suppose that $A$ is $2\times 2$ matrix that has eigenvalues $-1$ and $3$.
Then for each positive integer $n$ find $a_n$ and $b_n$ such that
\[A^{n+1}=a_nA+b_nI,\]
where $I$ is the $2\times 2$ identity matrix.
Add to solve later Let $V$ be a real vector space of all real sequences
\[(a_i)_{i=1}^{\infty}=(a_1, a_2, \dots).\]
Let $U$ be the subspace of $V$ consisting of all real sequences that satisfy the linear recurrence relation
\[a_{k+2}-5a_{k+1}+3a_{k}=0\]
for $k=1, 2, \dots$.
Let $T$ be the linear transformation from $U$ to $U$ defined by
\[T\big((a_1, a_2, \dots)\big)=(a_2, a_3, \dots). \]
Let $B=\{\mathbf{u}_1, \mathbf{u}_2\}$ be a basis of $U$, where
\begin{align*}
\mathbf{u}_1&=(1, 0, -3, -15, -66, \dots)\\
\mathbf{u}_2&=(0, 1, 5, 22, 95, \dots).
\end{align*}
Let $A$ be the matrix representation of the linear transformation $T: U \to U$ with respect to the basis $B$.
(a) Find the eigenvalues and eigenvectors of $T$.
(b) Use the result of (a), find a sequence $(a_i)_{i=1}^{\infty}$ satisfying the linear recurrence relation $a_{k+2}-5a_{k+1}+3a_{k}=0$ and the initial condition $a_1=1, a_2=1$.
(c) Find the formula for the sequences $(a_i)_{i=1}^{\infty}$ satisfying the linear recurrence relation $a_{k+2}-5a_{k+1}+3a_{k}=0$ and express it using $a_1, a_2$.
Add to solve later Let
\[A=\begin{bmatrix}
a & -1\\
1& 4
\end{bmatrix}\]
be a $2\times 2$ matrix, where $a$ is some real number.
Suppose that the matrix $A$ has an eigenvalue $3$.
(a) Determine the value of $a$.
(b) Does the matrix $A$ have eigenvalues other than $3$?
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 For a real number $0\leq \theta \leq \pi$, we define the real $3\times 3$ matrix $A$ by
\[A=\begin{bmatrix}
\cos\theta & -\sin\theta & 0 \\
\sin\theta &\cos\theta &0 \\
0 & 0 & 1
\end{bmatrix}.\]
(a) Find the determinant of the matrix $A$.
(b) Show that $A$ is an orthogonal matrix.
(c) Find the eigenvalues of $A$.
Add to solve later 
