## Problem 379

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)

## Problem 378

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.

## Problem 377

Let $A$ be an $n\times n$ idempotent complex matrix.
Then prove that $A$ is diagonalizable.

## Problem 376

(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 $\lambda=0$.

## Problem 375

Let $n>1$ be a positive integer. Let $V=M_{n\times n}(\C)$ be the vector space over the complex numbers $\C$ consisting of all complex $n\times n$ matrices. The dimension of $V$ is $n^2$.
Let $A \in V$ and consider the set
$S_A=\{I=A^0, A, A^2, \dots, A^{n^2-1}\}$ of $n^2$ elements.
Prove that the set $S_A$ cannot be a basis of the vector space $V$ for any $A\in V$.

## Problem 374

Let $A=\begin{bmatrix} a_0 & a_1 & \dots & a_{n-2} &a_{n-1} \\ a_{n-1} & a_0 & \dots & a_{n-3} & a_{n-2} \\ a_{n-2} & a_{n-1} & \dots & a_{n-4} & a_{n-3} \\ \vdots & \vdots & \dots & \vdots & \vdots \\ a_{2} & a_3 & \dots & a_{0} & a_{1}\\ a_{1} & a_2 & \dots & a_{n-1} & a_{0} \end{bmatrix}$ be a complex $n \times n$ matrix.
Such a matrix is called circulant matrix.
Then prove that the determinant of the circulant matrix $A$ is given by
$\det(A)=\prod_{k=0}^{n-1}(a_0+a_1\zeta^k+a_2 \zeta^{2k}+\cdots+a_{n-1}\zeta^{k(n-1)}),$ where $\zeta=e^{2 \pi i/n}$ is a primitive $n$-th root of unity.

## Problem 373

Let $A$ be a $3\times 3$ matrix. Suppose that $A$ has eigenvalues $2$ and $-1$, and suppose that $\mathbf{u}$ and $\mathbf{v}$ are eigenvectors corresponding to $2$ and $-1$, respectively, where
$\mathbf{u}=\begin{bmatrix} 1 \\ 0 \\ -1 \end{bmatrix} \text{ and } \mathbf{v}=\begin{bmatrix} 2 \\ 1 \\ 0 \end{bmatrix}.$ Then compute $A^5\mathbf{w}$, where
$\mathbf{w}=\begin{bmatrix} 7 \\ 2 \\ -3 \end{bmatrix}.$

## Problem 372

For each positive integer $n$, prove that the polynomial
$(x-1)(x-2)\cdots (x-n)-1$ is irreducible over the ring of integers $\Z$.

## Problem 371

Let $S$ be the subset of $\R^4$ consisting of vectors $\begin{bmatrix} x \\ y \\ z \\ w \end{bmatrix}$ satisfying
$2x+3y+5z+7w=0.$ Then prove that the set $S$ is a subspace of $\R^4$.

(Linear Algebra Exam Problem, The Ohio State University)

## Problem 370

Let $T: \R^2 \to \R^2$ be a linear transformation such that
$T\left(\, \begin{bmatrix} 1 \\ 1 \end{bmatrix} \,\right)=\begin{bmatrix} 4 \\ 1 \end{bmatrix}, T\left(\, \begin{bmatrix} 0 \\ 1 \end{bmatrix} \,\right)=\begin{bmatrix} 3 \\ 2 \end{bmatrix}.$ Then find the matrix $A$ such that $T(\mathbf{x})=A\mathbf{x}$ for every $\mathbf{x}\in \R^2$, and find the rank and nullity of $T$.

(The Ohio State University, Linear Algebra Exam Problem)

## Problem 369

Let $T:\R^3 \to \R^2$ be a linear transformation such that
$T(\mathbf{e}_1)=\begin{bmatrix} 1 \\ 0 \end{bmatrix}, T(\mathbf{e}_2)=\begin{bmatrix} 0 \\ 1 \end{bmatrix}, T(\mathbf{e}_3)=\begin{bmatrix} 1 \\ 0 \end{bmatrix},$ where $\mathbf{e}_1, \mathbf{e}_2, \mathbf{e}_3$ are the standard basis of $\R^3$.
Then find the rank and the nullity of $T$.

(The Ohio State University, Linear Algebra Exam Problem)

## Problem 368

Let $T$ be a linear transformation from $\R^3$ to $\R^2$ such that
$T\left(\, \begin{bmatrix} 0 \\ 1 \\ 0 \end{bmatrix}\,\right) =\begin{bmatrix} 1 \\ 2 \end{bmatrix} \text{ and }T\left(\, \begin{bmatrix} 0 \\ 1 \\ 1 \end{bmatrix}\,\right)=\begin{bmatrix} 0 \\ 1 \end{bmatrix}.$ Then find $T\left(\, \begin{bmatrix} 0 \\ 1 \\ 2 \end{bmatrix} \,\right)$.

(The Ohio State University, Linear Algebra Exam Problem)

## Problem 367

Let $P_2$ be the vector space of all polynomials of degree $2$ or less with real coefficients.
Let
$S=\{1+x+2x^2, \quad x+2x^2, \quad -1, \quad x^2\}$ be the set of four vectors in $P_2$.

Then find a basis of the subspace $\Span(S)$ among the vectors in $S$.

(Linear Algebra Exam Problem, the Ohio State University)

## Problem 366

Let $A=\begin{bmatrix} 1 & 0 & 1 \\ 0 &1 &0 \end{bmatrix}$.

(a) Find an orthonormal basis of the null space of $A$.

(b) Find the rank of $A$.

(c) Find an orthonormal basis of the row space of $A$.

(The Ohio State University, Linear Algebra Exam Problem)

## Problem 365

Let $f(x)=\sin^2(x)$, $g(x)=\cos^2(x)$, and $h(x)=1$. These are vectors in $C[-1, 1]$.
Determine whether the set $\{f(x), \, g(x), \, h(x)\}$ is linearly dependent or linearly independent.

(The Ohio State University, Linear Algebra Midterm Exam Problem)

## Problem 364

These are True or False problems.
For each of the following statements, determine if it contains a wrong information or not.

1. Let $A$ be a $5\times 3$ matrix. Then the range of $A$ is a subspace in $\R^3$.
2. The function $f(x)=x^2+1$ is not in the vector space $C[-1,1]$ because $f(0)=1\neq 0$.
3. Since we have $\sin(x+y)=\sin(x)+\sin(y)$, the function $\sin(x)$ is a linear transformation.
4. The set
$\left\{\, \begin{bmatrix} 1 \\ 0 \\ 0 \end{bmatrix}, \begin{bmatrix} 0 \\ 1 \\ 1 \end{bmatrix} \,\right\}$ is an orthonormal set.

(Linear Algebra Exam Problem, The Ohio State University)

## Problem 363

(a) Find all the eigenvalues and eigenvectors of the matrix
$A=\begin{bmatrix} 3 & -2\\ 6& -4 \end{bmatrix}.$

(b) Let
$A=\begin{bmatrix} 1 & 0 & 3 \\ 4 &5 &6 \\ 7 & 0 & 9 \end{bmatrix} \text{ and } B=\begin{bmatrix} 2 & 0 & 0 \\ 0 & 3 &0 \\ 0 & 0 & 4 \end{bmatrix}.$ Then find the value of
$\det(A^2B^{-1}A^{-2}B^2).$ (For part (b) without computation, you may assume that $A$ and $B$ are invertible matrices.)

## Problem 362

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.

## Problem 361

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 $A^{-1}$.

## Problem 360

Let $R$ be a commutative ring and let $I_1$ and $I_2$ be comaximal ideals. That is, we have
$I_1+I_2=R.$

Then show that for any positive integers $m$ and $n$, the ideals $I_1^m$ and $I_2^n$ are comaximal.