# Tagged: nonsingular

## Problem 698

Let $A$ and $B$ be $3\times 3$ matrices and let $C=A-2B$.
If
$A\begin{bmatrix} 1 \\ 3 \\ 5 \end{bmatrix}=B\begin{bmatrix} 2 \\ 6 \\ 10 \end{bmatrix},$ then is the matrix $C$ nonsingular? If so, prove it. Otherwise, explain why not.

## Problem 671

Determine whether the following matrices are nonsingular or not.

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

(b) $B=\begin{bmatrix} 2 & 1 & 2 \\ 1 &0 &1 \\ 4 & 1 & 4 \end{bmatrix}$.

## Problem 657

Suppose that $M, P$ are two $n \times n$ non-singular matrix. Prove that there is a matrix $N$ such that $MN = P$.

## Problem 509

Using the numbers appearing in
$\pi=3.1415926535897932384626433832795028841971693993751058209749\dots$ we construct the matrix $A=\begin{bmatrix} 3 & 14 &1592& 65358\\ 97932& 38462643& 38& 32\\ 7950& 2& 8841& 9716\\ 939937510& 5820& 974& 9 \end{bmatrix}.$

Prove that the matrix $A$ is nonsingular.

## 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 299

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

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

## 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.

## Problem 146

Let $A$ be an $n \times n$ nilpotent matrix, that is, $A^m=O$ for some positive integer $m$, where $O$ is the $n \times n$ zero matrix.

Prove that $A$ is a singular matrix and also prove that $I-A, I+A$ are both nonsingular matrices, where $I$ is the $n\times n$ identity matrix.

## Problem 26

In this problem, we will show that the concept of non-singularity of a matrix is equivalent to the concept of invertibility.
That is, we will prove that:

A matrix $A$ is nonsingular if and only if $A$ is invertible.

(a) Show that if $A$ is invertible, then $A$ is nonsingular.

(b) Let $A, B, C$ be $n\times n$ matrices such that $AB=C$.
Prove that if either $A$ or $B$ is singular, then so is $C$.

(c) Show that if $A$ is nonsingular, then $A$ is invertible.

## Problem 25

An $n \times n$ matrix $A$ is called nonsingular if the only solution of the equation $A \mathbf{x}=\mathbf{0}$ is the zero vector $\mathbf{x}=\mathbf{0}$.
Otherwise $A$ is called singular.

(a) Show that if $A$ and $B$ are $n\times n$ nonsingular matrices, then the product $AB$ is also nonsingular.

(b) Show that if $A$ is nonsingular, then the column vectors of $A$ are linearly independent.

(c) Show that an $n \times n$ matrix $A$ is nonsingular if and only if the equation $A\mathbf{x}=\mathbf{b}$ has a unique solution for any vector $\mathbf{b}\in \R^n$.

Restriction
Do not use the fact that a matrix is nonsingular if and only if the matrix is invertible.