# 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.

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