# Tagged: nonsingular matrix

## Problem 547

Let $A$ be an $n\times n$ nonsingular matrix with integer entries.

Prove that the inverse matrix $A^{-1}$ contains only integer entries if and only if $\det(A)=\pm 1$.

## 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 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 486

Determine whether there exists a nonsingular matrix $A$ if
$A^4=ABA^2+2A^3,$ where $B$ is the following matrix.
$B=\begin{bmatrix} -1 & 1 & -1 \\ 0 &-1 &0 \\ 2 & 1 & -4 \end{bmatrix}.$

If such a nonsingular matrix $A$ exists, find the inverse matrix $A^{-1}$.

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

## Problem 479

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

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

## Problem 477

Determine whether the matrix
$A=\begin{bmatrix} 1 & 4\\ 2 & 3 \end{bmatrix}$ is diagonalizable.

If so, find a nonsingular matrix $S$ and a diagonal matrix $D$ such that $S^{-1}AS=D$.

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

## Problem 468

Let $A$ be an $n\times n$ real skew-symmetric matrix.

(a) Prove that the matrices $I-A$ and $I+A$ are nonsingular.

(b) Prove that
$B=(I-A)(I+A)^{-1}$ is an orthogonal matrix.

## Problem 394

Determine the values of $x$ so that the matrix
$A=\begin{bmatrix} 1 & 1 & x \\ 1 &x &x \\ x & x & x \end{bmatrix}$ is invertible.
For those values of $x$, find the inverse matrix $A^{-1}$.

## 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 388

Let $A$ be $n\times n$ matrix and let $\lambda_1, \lambda_2, \dots, \lambda_n$ be all the eigenvalues of $A$. (Some of them may be the same.)

For each positive integer $k$, prove that $\lambda_1^k, \lambda_2^k, \dots, \lambda_n^k$ are all the eigenvalues of $A^k$.

## 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 319

Let $A, B$, and $C$ be $n \times n$ matrices and $I$ be the $n\times n$ identity matrix.
Prove the following statements.

(a) If $A$ is similar to $B$, then $B$ is similar to $A$.

(b) $A$ is similar to itself.

(c) If $A$ is similar to $B$ and $B$ is similar to $C$, then $A$ is similar to $C$.

(d) If $A$ is similar to the identity matrix $I$, then $A=I$.

(e) If $A$ or $B$ is nonsingular, then $AB$ is similar to $BA$.

(f) If $A$ is similar to $B$, then $A^k$ is similar to $B^k$ for any positive integer $k$.

## 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 284

Let $\mathbf{v}_1$ and $\mathbf{v}_2$ be $2$-dimensional vectors and let $A$ be a $2\times 2$ matrix.

(a) Show that if $\mathbf{v}_1, \mathbf{v}_2$ are linearly dependent vectors, then the vectors $A\mathbf{v}_1, A\mathbf{v}_2$ are also linearly dependent.

(b) If $\mathbf{v}_1, \mathbf{v}_2$ are linearly independent vectors, can we conclude that the vectors $A\mathbf{v}_1, A\mathbf{v}_2$ are also linearly independent?

(c) If $\mathbf{v}_1, \mathbf{v}_2$ are linearly independent vectors and $A$ is nonsingular, then show that the vectors $A\mathbf{v}_1, A\mathbf{v}_2$ are also linearly independent.

## Problem 280

Determine whether there exists a nonsingular matrix $A$ if
$A^2=AB+2A,$ where $B$ is the following matrix.
If such a nonsingular matrix $A$ exists, find the inverse matrix $A^{-1}$.

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

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

## 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 271

Let $A$ be an $n\times n$ singular matrix.
Then prove that there exists a nonzero $n\times n$ matrix $B$ such that
$AB=O,$ where $O$ is the $n\times n$ zero matrix.

## Problem 266

Let $A$ be an $n \times n$ matrix satisfying
$A^2+c_1A+c_0I=O,$ where $c_0, c_1$ are scalars, $I$ is the $n\times n$ identity matrix, and $O$ is the $n\times n$ zero matrix.

Prove that if $c_0\neq 0$, then the matrix $A$ is invertible (nonsingular).
How about the converse? Namely, is it true that if $c_0=0$, then the matrix $A$ is not invertible?

## Problem 213

Let $A, B$ be matrices. Show that if $A$ is diagonalizable and if $B$ is similar to $A$, then $B$ is diagonalizable.

## Problem 211

In this post, we explain how to diagonalize a matrix if it is diagonalizable.

As an example, we solve the following problem.

Diagonalize the matrix
$A=\begin{bmatrix} 4 & -3 & -3 \\ 3 &-2 &-3 \\ -1 & 1 & 2 \end{bmatrix}$ by finding a nonsingular matrix $S$ and a diagonal matrix $D$ such that $S^{-1}AS=D$.

(Update 10/15/2017. A new example problem was added.)