# Determine Whether There Exists a Nonsingular Matrix Satisfying $A^4=ABA^2+2A^3$

## 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)

## Solution.

Assume that $A$ is nonsingular.
Then the inverse matrix $A^{-1}$ exists and we have
\begin{align*}
A&=A^{-1}A^4A^{-2}\\
&=A^{-1}(ABA^2+2A^3)A^{-2}\\
&=A^{-1}ABA^2 A^{-2}+2A^{-1}A^3A^{-2}\\
&=B+2I,
\end{align*}
where $I$ is the $3\times 3$ identity matrix.

Thus we have
\begin{align*}
A=B+2I=\begin{bmatrix}
1 & 1 & -1 \\
0 &1 &0 \\
2 & 1 & -2
\end{bmatrix}.
\end{align*}

We compute the determinant of $A$ by the second row cofactor expansion as follows.
\begin{align*}
\det(A)=\begin{vmatrix}
1 & 1 & -1 \\
0 &1 &0 \\
2 & 1 & -2
\end{vmatrix}
=\begin{vmatrix}
1 & -1\\
2& -2
\end{vmatrix}=0.
\end{align*}
This contradicts the assumption that $A$ is nonsingular.

Therefore, there is no nonsingular matrix $A$ satisfying the relation $A^4=ABA^2+2A^3$.

## Final Exam Problems and Solution. (Linear Algebra Math 2568 at the Ohio State University)

This problem is one of the final exam problems of Linear Algebra course at the Ohio State University (Math 2568).

The other problems can be found from the links below.

### More from my site

• Maximize the Dimension of the Null Space of $A-aI$ Let $A=\begin{bmatrix} 5 & 2 & -1 \\ 2 &2 &2 \\ -1 & 2 & 5 \end{bmatrix}.$ Pick your favorite number $a$. Find the dimension of the null space of the matrix $A-aI$, where $I$ is the $3\times 3$ identity matrix. Your score of this problem is equal to that […]
• The Product of Two Nonsingular Matrices is Nonsingular 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)   Definition (Nonsingular Matrix) An $n\times n$ matrix is called nonsingular if the […]
• Quiz 4: Inverse Matrix/ Nonsingular Matrix Satisfying a Relation (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 […]
• Diagonalize a 2 by 2 Matrix if Diagonalizable 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 […]
• Find the Nullity of the Matrix $A+I$ if Eigenvalues are $1, 2, 3, 4, 5$ 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 […]
• Given the Characteristic Polynomial, Find the Rank of the Matrix Let $A$ be a square matrix and its characteristic polynomial is give by $p(t)=(t-1)^3(t-2)^2(t-3)^4(t-4).$ Find the rank of $A$. (The Ohio State University, Linear Algebra Final Exam Problem)   Solution. Note that the degree of the characteristic polynomial […]
• Find Values of $a, b, c$ such that the Given Matrix is Diagonalizable For which values of constants $a, b$ and $c$ is the matrix $A=\begin{bmatrix} 7 & a & b \\ 0 &2 &c \\ 0 & 0 & 3 \end{bmatrix}.$ diagonalizable? (The Ohio State University, Linear Algebra Final Exam Problem)   Solution. Note that […]
• Find Values of $h$ so that the Given Vectors are Linearly Independent 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 […]

### 3 Responses

1. 06/28/2017

[…] Determine Whether There Exists a Nonsingular Matrix Satisfying $A^4=ABA^2+2A^3$ […]

2. 08/02/2017

[…] Determine Whether There Exists a Nonsingular Matrix Satisfying $A^4=ABA^2+2A^3$ […]

3. 09/13/2017

[…] Determine Whether There Exists a Nonsingular Matrix Satisfying $A^4=ABA^2+2A^3$ […]

##### Compute $A^{10}\mathbf{v}$ Using Eigenvalues and Eigenvectors of the Matrix $A$

Let \[A=\begin{bmatrix} 1 & -14 & 4 \\ -1 &6 &-2 \\ -2 & 24 & -7 \end{bmatrix} \quad \text{...

Close