# Find a Nonsingular Matrix $A$ satisfying $3A=A^2+AB$

## Problem 699

(a) Find a $3\times 3$ nonsingular matrix $A$ satisfying $3A=A^2+AB$, where $B=\begin{bmatrix} 2 & 0 & -1 \\ 0 &2 &-1 \\ -1 & 0 & 1 \end{bmatrix}.$

(b) Find the inverse matrix of $A$.

## Solution

### (a) Find a $3\times 3$ nonsingular matrix $A$.

Assume that $A$ is nonsingular. Then the inverse matrix $A^{-1}$ exists.
Multiplying the given equality by $A^{-1}$ on the left, we obtain
$3A^{-1}A=A^{-1}(A^2+AB)=A^{-1}A^2+A^{-1}AB=A+IB=A+B.$ Note that the left most term is equal to $3I$.
Hence, we have $3I=A+B$. Solving for $A$, we have
$A=3I-B=\begin{bmatrix} 3 & 0 & 0 \\ 0 &3 &0 \\ 0 & 0 & 3 \end{bmatrix}-\begin{bmatrix} 2 & 0 & -1 \\ 0 &2 &-1 \\ -1 & 0 & 1 \end{bmatrix}=\begin{bmatrix} 1 & 0 & 1 \\ 0 &1 &1 \\ 1 & 0 & 2 \end{bmatrix}.$

As we see in part (b), this matrix is actually invertible.

### (b) Find the inverse matrix of $A$.

To find the inverse matrix of $A$, we reduce the augmented matrix $[A\mid I]$ as follows:
\begin{align*}
[A\mid I]= \left[\begin{array}{rrr|rrr}
1 & 0 & 1 & 1 &0 & 0 \\
0 & 1 & 1 & 0 & 1 & 0 \\
1 & 0 & 2 & 0 & 0 & 1 \\
\end{array} \right] \xrightarrow{R_3-R_2}
\left[\begin{array}{rrr|rrr}
1 & 0 & 1 & 1 &0 & 0 \\
0 & 1 & 1 & 0 & 1 & 0 \\
0 & 0 & 1 & -1 & 0 & 1 \\
\end{array} \right]\6pt] \xrightarrow[R_2-R_3]{R_1-R_3} \left[\begin{array}{rrr|rrr} 1 & 0 & 0 & 2 &0 & -1 \\ 0 & 1 & 0 & 1 & 1 & -1 \\ 0 & 0 & 1 & -1 & 0 & 1 \\ \end{array} \right]. \end{align*} The left 3 by 3 part is now the identity matrix. So the inverse matrix is given by the right 3 by 3 part: \[A^{-1}=\begin{bmatrix} 2 & 0 & -1 \\ 1 &1 &-1 \\ -1 & 0 & 1 \end{bmatrix}.

## Common Mistake

This is a midterm exam problem of Lienar Algebra at the Ohio State University.

One common mistake is: $A=3-B$. Don’t forget the identity matrix $I$. $3$ is a number but $3I$ is a matrix.
Also $3I$ is the 3 by 3 matrix whose diagonal entries are 3 and 0 elsewhere. Note that $3I$ is not the 3 by3 matrix whose entries are all 3.

### More from my site

• 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 […]
• Find the Inverse Matrix of a $3\times 3$ Matrix if Exists 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 […]
• Using Properties of Inverse Matrices, Simplify the Expression Let $A, B, C$ be $n\times n$ invertible matrices. When you simplify the expression $C^{-1}(AB^{-1})^{-1}(CA^{-1})^{-1}C^2,$ which matrix do you get? (a) $A$ (b) $C^{-1}A^{-1}BC^{-1}AC^2$ (c) $B$ (d) $C^2$ (e) $C^{-1}BC$ (f) $C$   Solution. In this problem, we […]
• True or False Problems on Midterm Exam 1 at OSU Spring 2018 The following problems are True or False. Let $A$ and $B$ be $n\times n$ matrices. (a) If $AB=B$, then $B$ is the identity matrix. (b) If the coefficient matrix $A$ of the system $A\mathbf{x}=\mathbf{b}$ is invertible, then the system has infinitely many solutions. (c) If $A$ […]
• 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 a Nonsingular Matrix Satisfying Some Relation 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 & […] • Solve a System by the Inverse Matrix and Compute A^{2017}\mathbf{x} Let A be the coefficient matrix of the system of linear equations \begin{align*} -x_1-2x_2&=1\\ 2x_1+3x_2&=-1. \end{align*} (a) Solve the system by finding the inverse matrix A^{-1}. (b) Let \mathbf{x}=\begin{bmatrix} x_1 \\ x_2 \end{bmatrix} be the solution […] • Determine Whether There Exists a Nonsingular Matrix Satisfying A^4=ABA^2+2A^3 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 […]

#### You may also like...

This site uses Akismet to reduce spam. Learn how your comment data is processed.

##### Determine whether the Matrix is Nonsingular from the Given Relation

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...

Close