# Tagged: quiz

## Problem 500

10 questions about nonsingular matrices, invertible matrices, and linearly independent vectors.

The quiz is designed to test your understanding of the basic properties of these topics.

You can take the quiz as many times as you like.

The solutions will be given after completing all the 10 problems.
Click the View question button to see the solutions.

## Problem 386

Find all eigenvalues of the matrix
$A=\begin{bmatrix} 0 & i & i & i \\ i &0 & i & i \\ i & i & 0 & i \\ i & i & i & 0 \end{bmatrix},$ where $i=\sqrt{-1}$. For each eigenvalue of $A$, determine its algebraic multiplicity and geometric multiplicity.

## Problem 385

Let
$A=\begin{bmatrix} 2 & -1 & -1 \\ -1 &2 &-1 \\ -1 & -1 & 2 \end{bmatrix}.$ Determine whether the matrix $A$ is diagonalizable. If it is diagonalizable, then diagonalize $A$.
That is, find a nonsingular matrix $S$ and a diagonal matrix $D$ such that $S^{-1}AS=D$.

## 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 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 349

Let $V$ be the vector space of all $2\times 2$ real matrices.
Let $S=\{A_1, A_2, A_3, A_4\}$, where
$A_1=\begin{bmatrix} 1 & 2\\ -1& 3 \end{bmatrix}, A_2=\begin{bmatrix} 0 & -1\\ 1& 4 \end{bmatrix}, A_3=\begin{bmatrix} -1 & 0\\ 1& -10 \end{bmatrix}, A_4=\begin{bmatrix} 3 & 7\\ -2& 6 \end{bmatrix}.$ Then find a basis for the span $\Span(S)$.

## Problem 328

(a) Let $C[-1,1]$ be the vector space over $\R$ of all real-valued continuous functions defined on the interval $[-1, 1]$.
Consider the subset $F$ of $C[-1, 1]$ defined by
$F=\{ f(x)\in C[-1, 1] \mid f(0) \text{ is an integer}\}.$ Prove or disprove that $F$ is a subspace of $C[-1, 1]$.

(b) Let $n$ be a positive integer.
An $n\times n$ matrix $A$ is called skew-symmetric if $A^{\trans}=-A$.
Let $M_{n\times n}$ be the vector space over $\R$ of all $n\times n$ real matrices.
Consider the subset $W$ of $M_{n\times n}$ defined by
$W=\{A\in M_{n\times n} \mid A \text{ is skew-symmetric}\}.$ Prove or disprove that $W$ is a subspace of $M_{n\times n}$.

## Problem 320

(a) Let $A=\begin{bmatrix} 1 & 3 & 0 & 0 \\ 1 &3 & 1 & 2 \\ 1 & 3 & 1 & 2 \end{bmatrix}$.
Find a basis for the range $\calR(A)$ of $A$ that consists of columns of $A$.

(b) Find the rank and nullity of the matrix $A$ in part (a).

## Problem 313

(a) Let $A=\begin{bmatrix} 1 & 2 & 1 \\ 3 &6 &4 \end{bmatrix}$ and let
$\mathbf{a}=\begin{bmatrix} -3 \\ 1 \\ 1 \end{bmatrix}, \qquad \mathbf{b}=\begin{bmatrix} -2 \\ 1 \\ 0 \end{bmatrix}, \qquad \mathbf{c}=\begin{bmatrix} 1 \\ 1 \end{bmatrix}.$ For each of the vectors $\mathbf{a}, \mathbf{b}, \mathbf{c}$, determine whether the vector is in the null space $\calN(A)$. Do the same for the range $\calR(A)$.

(b) Find a basis of the null space of the matrix $B=\begin{bmatrix} 1 & 1 & 2 \\ -2 &-2 &-4 \end{bmatrix}$.

## Problem 304

Problem 1 Let $W$ be the subset of the $3$-dimensional vector space $\R^3$ defined by
$W=\left\{ \mathbf{x}=\begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix}\in \R^3 \quad \middle| \quad 2x_1x_2=x_3 \right\}.$

(a) Which of the following vectors are in the subset $W$? Choose all vectors that belong to $W$.
$(1) \begin{bmatrix} 0 \\ 0 \\ 0 \end{bmatrix} \qquad(2) \begin{bmatrix} 1 \\ 2 \\ 2 \end{bmatrix} \qquad(3)\begin{bmatrix} 3 \\ 0 \\ 0 \end{bmatrix} \qquad(4) \begin{bmatrix} 0 \\ 0 \end{bmatrix} \qquad(5) \begin{bmatrix} 1 & 2 & 4 \\ 1 &2 &4 \end{bmatrix} \qquad(6) \begin{bmatrix} 1 \\ -1 \\ -2 \end{bmatrix}.$

(b) Determine whether $W$ is a subspace of $\R^3$ or not.

Problem 2 Let $W$ be the subset of $\R^3$ defined by
$W=\left\{ \mathbf{x}=\begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix} \in \R^3 \quad \middle| \quad x_1=3x_2 \text{ and } x_3=0 \right\}.$ Determine whether the subset $W$ is a subspace of $\R^3$ or not.

## Problem 289

(a) Find the inverse matrix of
$A=\begin{bmatrix} 1 & 0 & 1 \\ 1 &0 &0 \\ 2 & 1 & 1 \end{bmatrix}$ if it exists. If you think there is no inverse matrix of $A$, then give a reason.

(b) Find a nonsingular $2\times 2$ matrix $A$ such that
$A^3=A^2B-3A^2,$ where
$B=\begin{bmatrix} 4 & 1\\ 2& 6 \end{bmatrix}.$ Verify that the matrix $A$ you obtained is actually a nonsingular matrix.

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

## Problem 281

(a) For what value(s) of $a$ is the following set $S$ linearly dependent?
$S=\left \{\,\begin{bmatrix} 1 \\ 2 \\ 3 \\ a \end{bmatrix}, \begin{bmatrix} a \\ 0 \\ -1 \\ 2 \end{bmatrix}, \begin{bmatrix} 0 \\ 0 \\ a^2 \\ 7 \end{bmatrix}, \begin{bmatrix} 1 \\ a \\ 1 \\ 1 \end{bmatrix}, \begin{bmatrix} 2 \\ -2 \\ 3 \\ a^3 \end{bmatrix} \, \right\}.$

(b) Let $\{\mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3\}$ be a set of nonzero vectors in $\R^m$ such that the dot product
$\mathbf{v}_i\cdot \mathbf{v}_j=0$ when $i\neq j$.
Prove that the set is linearly independent.

## Problem 273

(a) The given matrix is the augmented matrix for a system of linear equations.
Give the vector form for the general solution.
$\left[\begin{array}{rrrrr|r} 1 & 0 & -1 & 0 &-2 & 0 \\ 0 & 1 & 2 & 0 & -1 & 0 \\ 0 & 0 & 0 & 1 & 1 & 0 \\ \end{array} \right].$

(b) Let
$A=\begin{bmatrix} 1 & 2 & 3 \\ 4 &5 &6 \end{bmatrix}, B=\begin{bmatrix} 1 & 0 & 1 \\ 0 &1 &0 \end{bmatrix}, C=\begin{bmatrix} 1 & 2\\ 0& 6 \end{bmatrix}, \mathbf{v}=\begin{bmatrix} 0 \\ 1 \\ 0 \end{bmatrix}.$ Then compute and simplify the following expression.
$\mathbf{v}^{\trans}\left( A^{\trans}-(A-B)^{\trans}\right)C.$

## Problem 262

(a) Solve the following system by transforming the augmented matrix to reduced echelon form (Gauss-Jordan elimination). Indicate the elementary row operations you performed.
\begin{align*}
x_1+x_2-x_5&=1\\
x_2+2x_3+x_4+3x_5&=1\\
x_1-x_3+x_4+x_5&=0
\end{align*}

(b) Determine all possibilities for the solution set of a homogeneous system of $2$ equations in $2$ unknowns that has a solution $x_1=1, x_2=5$.

## Problem 96

Let $A$ and $B$ be $2\times 2$ matrices.

Prove or find a counterexample for the statement that $(A-B)(A+B)=A^2-B^2$.

## Problem 93

4 multiple choice questions about possibilities for the solution set of a homogeneous system of linear equations.

The solutions will be given after completing all problems.