Show that the matrix $A=\begin{bmatrix}
1 & \alpha\\
0& 1
\end{bmatrix}$, where $\alpha$ is an element of a field $F$ of characteristic $p>0$ satisfies $A^p=I$ and the matrix is not diagonalizable over $F$ if $\alpha \neq 0$. Read solution

Prove that the polynomial $x^p-2$ for a prime number $p$ is irreducible over the field $\Q(\zeta_p)$, where $\zeta_p$ is a primitive $p$th root of unity.

A complex number $z$ is called algebraic number (respectively, algebraic integer) if $z$ is a root of a monic polynomial with rational (respectively, integer) coefficients.

Prove that $z \in \C$ is an algebraic number (resp. algebraic integer) if and only if $z$ is an eigenvalue of a matrix with rational (resp. integer) entries.

Then prove that the characteristic polynomial $\det(xI-A)$ of $A$ is the polynomial $p(x)$.
The matrix is called the companion matrix of the polynomial $p(x)$.

Let $V$ be a finite dimensional vector space over a field $K$ and let $\End (V)$ be the vector space of linear transformations from $V$ to $V$.
Let $\mathbf{v}_1, \mathbf{v}_2, \dots, \mathbf{v}_n$ be a basis for $V$.
Show that the map $\phi:\End (V) \to V^{\oplus n}$ defined by $f\mapsto (f(\mathbf{v}_1), \dots, f(\mathbf{v}_n))$ is an isomorphism.
Here $V^{\oplus n}=V\oplus \dots \oplus V$, the direct sum of $n$ copies of $V$. Read solution

Let $E_{ij}$ denote the $n \times n$ matrix whose $(i,j)$-entry is $1$ and zero elsewhere.

(a) Show that $V$ is a subspace of the vector space $M_n$ over $\C$ of all $n\times n$ matrices. (You may assume without a proof that $M_n$ is a vector space.)

(b) Show that matrices
\[E_{11}-E_{22}, \, E_{22}-E_{33}, \, \dots,\, E_{n-1\, n-1}-E_{nn}\]
are a basis for the vector space $V$.

Determine whether the following sentence is True or False.

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Question 1 of 3

1. Question

True or False. A linear system of four equations in three unknowns is always inconsistent.

Correct

Good! For example, the homogeneous system
\[\left\{
\begin{array}{c}
x+y+z=0 \\
2x+2y+2z=0 \\
3x+3y+3z=0
\end{array}
\right.
\]
has the solution $(x,y,z)=(0,0,0)$. So the system is consistent.

Incorrect

the homogeneous system
\[\left\{
\begin{array}{c}
x+y+z=0 \\
2x+2y+2z=0 \\
3x+3y+3z=0
\end{array}
\right.
\]
has the solution $(x,y,z)=(0,0,0)$. So the system is consistent.

Question 2 of 3

2. Question

True or False. A linear system with fewer equations than unknowns must have infinitely many solutions.

Correct

Good! For example, consider the system of one equation with two unknowns
\[0x+0y=1.\]
This system has no solution at all.

Incorrect

For example, consider the system of one equation with two unknowns
\[0x+0y=1.\]
This system has no solution at all.

Question 3 of 3

3. Question

True or False. If the system $A\mathbf{x}=\mathbf{b}$ has a unique solution, then $A$ must be a square matrix.

Correct

Good! For example, consider the matrix $A=\begin{bmatrix}
1 \\
1
\end{bmatrix}$. Then the system
\[\begin{bmatrix}
1 \\
1
\end{bmatrix}[x]=\begin{bmatrix}
0 \\
0
\end{bmatrix}\]
has the unique solution $x=0$ but $A$ is not a square matrix.

Incorrect

For example, consider the matrix $A=\begin{bmatrix}
1 \\
1
\end{bmatrix}$. Then the system
\[\begin{bmatrix}
1 \\
1
\end{bmatrix}[x]=\begin{bmatrix}
0 \\
0
\end{bmatrix}\]
has the unique solution $x=0$ but $A$ is not a square matrix.

A square matrix $A$ is called nilpotent if there exists a positive integer $k$ such that $A^k=O$, where $O$ is the zero matrix.

(a) If $A$ is a nilpotent $n \times n$ matrix and $B$ is an $n\times n$ matrix such that $AB=BA$. Show that the product $AB$ is nilpotent.

(b) Let $P$ be an invertible $n \times n$ matrix and let $N$ be a nilpotent $n\times n$ matrix. Is the product $PN$ nilpotent? If so, prove it. If not, give a counterexample.

Let A be the matrix
\[\begin{bmatrix}
1 & -1 & 0 \\
0 &1 &-1 \\
0 & 0 & 1
\end{bmatrix}.\]
Is the matrix $A$ invertible? If not, then explain why it isn’t invertible. If so, then find the inverse.

Let $\Q$ denote the set of rational numbers (i.e., fractions of integers). Let $V$ denote the set of the form $x+y \sqrt{2}$ where $x,y \in \Q$. You may take for granted that the set $V$ is a vector space over the field $\Q$.

(a) Show that $B=\{1, \sqrt{2}\}$ is a basis for the vector space $V$ over $\Q$.

(b) Let $\alpha=a+b\sqrt{2} \in V$, and let $T_{\alpha}: V \to V$ be the map defined by
\[ T_{\alpha}(x+y\sqrt{2}):=(ax+2by)+(ay+bx)\sqrt{2}\in V\]
for any $x+y\sqrt{2} \in V$.
Show that $T_{\alpha}$ is a linear transformation.

(c) Let $\begin{bmatrix}
x \\
y
\end{bmatrix}_B=x+y \sqrt{2}$.
Find the matrix $T_B$ such that
\[ T_{\alpha} (x+y \sqrt{2})=\left( T_B\begin{bmatrix}
x \\
y
\end{bmatrix}\right)_B,\]
and compute $\det T_B$.

(a) Let $A=(a_{ij})$ be an $n\times n$ matrix. Suppose that the entries of the matrix $A$ satisfy the following relation.
\[|a_{ii}|>|a_{i1}|+\cdots +|a_{i\,i-1}|+|a_{i \, i+1}|+\cdots +|a_{in}|\]
for all $1 \leq i \leq n$.
Show that the matrix $A$ is nonsingular.

(b) Let $B=(b_{ij})$ be an $n \times n$ matrix whose entries satisfy the relation
\[ |b_{i\,i}|=1 \hspace{0.5cm} \text{ and }\hspace{0.5cm} |b_{ij}|<\frac{1}{n-1}\]
for all $i$ and $j$ with $i \neq j$.
Prove that the matrix $B$ is nonsingular.

(c)
Determine whether the following matrix is nonsingular or not.
\[C=\begin{bmatrix}
\pi & e & e^2/2\pi^2 \\[5 pt]
e^2/2\pi^2 &\pi &e \\[5pt]
e & e^2/2\pi^2 & \pi
\end{bmatrix},\]
where $\pi=3.14159\dots$, and $e=2.71828\dots$ is Euler’s number (or Napier’s constant).