# Tagged: singular matrix

## Problem 669

(a) Suppose that a $3\times 3$ system of linear equations is inconsistent. Is the coefficient matrix of the system nonsingular?

(b) Suppose that a $3\times 3$ homogeneous system of linear equations has a solution $x_1=0, x_2=-3, x_3=5$. Is the coefficient matrix of the system nonsingular?

(c) Let $A$ be a $4\times 4$ matrix and let
$\mathbf{v}=\begin{bmatrix} 1 \\ 2 \\ 3 \\ 4 \end{bmatrix} \text{ and } \mathbf{w}=\begin{bmatrix} 4 \\ 3 \\ 2 \\ 1 \end{bmatrix}.$ Suppose that we have $A\mathbf{v}=A\mathbf{w}$. Is the matrix $A$ nonsingular?

## Problem 655

Consider the matrix $M = \begin{bmatrix} 1 & 4 \\ 3 & 12 \end{bmatrix}$.

(a) Show that $M$ is singular.

(b) Find a non-zero vector $\mathbf{v}$ such that $M \mathbf{v} = \mathbf{0}$, where $\mathbf{0}$ is the $2$-dimensional zero vector.

## Problem 622

Let $A$ be an $n\times n$ matrix. Suppose that the sum of elements in each row of $A$ is zero.

Then prove that the matrix $A$ is singular.

## Problem 596

Let
$A=\begin{bmatrix} a & b\\ -b& a \end{bmatrix}$ be a $2\times 2$ matrix, where $a, b$ are real numbers.
Suppose that $b\neq 0$.

Prove that the matrix $A$ does not have real eigenvalues.

## Problem 561

Let $A$ be a singular $n\times n$ matrix.
Let
$\mathbf{e}_1=\begin{bmatrix} 1 \\ 0 \\ \vdots \\ 0 \end{bmatrix}, \mathbf{e}_2=\begin{bmatrix} 0 \\ 1 \\ \vdots \\ 0 \end{bmatrix}, \dots, \mathbf{e}_n=\begin{bmatrix} 0 \\ 0 \\ \vdots \\ 1 \end{bmatrix}$ be unit vectors in $\R^n$.

Prove that at least one of the following matrix equations
$A\mathbf{x}=\mathbf{e}_i$ for $i=1,2,\dots, n$, must have no solution $\mathbf{x}\in \R^n$.

## Problem 560

Let $A$ be an $n\times (n-1)$ matrix and let $\mathbf{b}$ be an $(n-1)$-dimensional vector.
Then the product $A\mathbf{b}$ is an $n$-dimensional vector.
Set the $n\times n$ matrix $B=[A_1, A_2, \dots, A_{n-1}, A\mathbf{b}]$, where $A_i$ is the $i$-th column vector of $A$.

Prove that $B$ is a singular matrix for any choice of $\mathbf{b}$.

## Problem 552

For each of the following $3\times 3$ matrices $A$, determine whether $A$ is invertible and find the inverse $A^{-1}$ if exists by computing the augmented matrix $[A|I]$, where $I$ is the $3\times 3$ identity matrix.

(a) $A=\begin{bmatrix} 1 & 3 & -2 \\ 2 &3 &0 \\ 0 & 1 & -1 \end{bmatrix}$

(b) $A=\begin{bmatrix} 1 & 0 & 2 \\ -1 &-3 &2 \\ 3 & 6 & -2 \end{bmatrix}$.

## Problem 505

Let $A$ be a singular $2\times 2$ matrix such that $\tr(A)\neq -1$ and let $I$ be the $2\times 2$ identity matrix.
Then prove that the inverse matrix of the matrix $I+A$ is given by the following formula:
$(I+A)^{-1}=I-\frac{1}{1+\tr(A)}A.$

Using the formula, calculate the inverse matrix of $\begin{bmatrix} 2 & 1\\ 1& 2 \end{bmatrix}$.

## Problem 473

Prove that the determinant of an $n\times n$ skew-symmetric matrix is zero if $n$ is odd.

## Problem 452

Let $A$ be an $n\times n$ complex matrix.
Let $S$ be an invertible matrix.

(a) If $SAS^{-1}=\lambda A$ for some complex number $\lambda$, then prove that either $\lambda^n=1$ or $A$ is a singular matrix.

(b) If $n$ is odd and $SAS^{-1}=-A$, then prove that $0$ is an eigenvalue of $A$.

(c) Suppose that all the eigenvalues of $A$ are integers and $\det(A) > 0$. If $n$ is odd and $SAS^{-1}=A^{-1}$, then prove that $1$ is an eigenvalue of $A$.

## Problem 406

Let $A$ be an $n\times n$ matrix. Suppose that $\mathbf{y}$ is a nonzero row vector such that
$\mathbf{y}A=\mathbf{y}.$ (Here a row vector means a $1\times n$ matrix.)
Prove that there is a nonzero column vector $\mathbf{x}$ such that
$A\mathbf{x}=\mathbf{x}.$ (Here a column vector means an $n \times 1$ matrix.)

## Problem 393

(a) Let $A$ be a $6\times 6$ matrix and suppose that $A$ can be written as
$A=BC,$ where $B$ is a $6\times 5$ matrix and $C$ is a $5\times 6$ matrix.

Prove that the matrix $A$ cannot be invertible.

(b) Let $A$ be a $2\times 2$ matrix and suppose that $A$ can be written as
$A=BC,$ where $B$ is a $2\times 3$ matrix and $C$ is a $3\times 2$ matrix.

Can the matrix $A$ be invertible?

## Problem 387

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)

## Problem 330

Let $V$ be the vector space of all $n\times n$ real matrices.
Let us fix a matrix $A\in V$.
Define a map $T: V\to V$ by
$T(X)=AX-XA$ for each $X\in V$.

(a) Prove that $T:V\to V$ is a linear transformation.

(b) Let $B$ be a basis of $V$. Let $P$ be the matrix representation of $T$ with respect to $B$. Find the determinant of $P$.

## Problem 301

Let $A$ be a $3\times 3$ singular matrix.

Then show that there exists a nonzero $3\times 3$ matrix $B$ such that
$AB=O,$ where $O$ is the $3\times 3$ zero matrix.

## Problem 279

Determine conditions on the scalars $a, b$ so that the following set $S$ of vectors is linearly dependent.
\begin{align*}
S=\{\mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3\},
\end{align*}
where
$\mathbf{v}_1=\begin{bmatrix} 1 \\ 3 \\ 1 \end{bmatrix}, \mathbf{v}_2=\begin{bmatrix} 1 \\ a \\ 4 \end{bmatrix}, \mathbf{v}_3=\begin{bmatrix} 0 \\ 2 \\ b \end{bmatrix}.$

## Problem 194

Find the value(s) of $h$ for which the following set of vectors
$\left \{ \mathbf{v}_1=\begin{bmatrix} 1 \\ 0 \\ 0 \end{bmatrix}, \mathbf{v}_2=\begin{bmatrix} h \\ 1 \\ -h \end{bmatrix}, \mathbf{v}_3=\begin{bmatrix} 1 \\ 2h \\ 3h+1 \end{bmatrix}\right\}$ is linearly independent.

(Boston College, Linear Algebra Midterm Exam Sample Problem)

## Problem 193

Let $A$ be a $3 \times 3$ matrix.
Let $\mathbf{x}, \mathbf{y}, \mathbf{z}$ are linearly independent $3$-dimensional vectors. Suppose that we have
$A\mathbf{x}=\begin{bmatrix} 1 \\ 0 \\ 1 \end{bmatrix}, A\mathbf{y}=\begin{bmatrix} 0 \\ 1 \\ 0 \end{bmatrix}, A\mathbf{z}=\begin{bmatrix} 1 \\ 1 \\ 1 \end{bmatrix}.$

Then find the value of the determinant of the matrix $A$.

## Problem 169

Find all the values of $x$ so that the following matrix $A$ is a singular matrix.
$A=\begin{bmatrix} x & x^2 & 1 \\ 2 &3 &1 \\ 0 & -1 & 1 \end{bmatrix}.$

## Problem 168

Let
$A=\begin{bmatrix} 1 & -x & 0 & 0 \\ 0 &1 & -x & 0 \\ 0 & 0 & 1 & -x \\ 0 & 1 & 0 & -1 \end{bmatrix}$ be a $4\times 4$ matrix. Find all values of $x$ so that the matrix $A$ is singular.