# Find the Inverse Matrix of a $3\times 3$ Matrix if Exists

## 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*)

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## Solution.

To check whether the matrix $A$ has the inverse matrix and to find the inverse matrix if exist at once, we consider the augmented matrix $[A\mid I]$, where $I$ is the $3\times 3$ identity matrix.

We apply the elementary row operations as follows.

\begin{align*}

[A\mid I] &= \left[\begin{array}{rrr|rrr}

1 & 1 & 2 & 1 &0 & 0 \\

0 & 0 & 1 & 0 & 1 & 0 \\

1 & 0 & 1 & 0 & 0 & 1 \\

\end{array} \right]
\xrightarrow{R_3-R_1}

\left[\begin{array}{rrr|rrr}

1 & 1 & 2 & 1 &0 & 0 \\

0 & 0 & 1 & 0 & 1 & 0 \\

0 & -1 & -1 & -1 & 0 & 1 \\

\end{array} \right]\\[10pt]
&\xrightarrow{-R_3}

\left[\begin{array}{rrr|rrr}

1 & 1 & 2 & 1 &0 & 0 \\

0 & 0 & 1 & 0 & 1 & 0 \\

0 & 1 & 1 & 1 & 0 & -1 \\

\end{array} \right]
\xrightarrow{R_2 \leftrightarrow R_3}

\left[\begin{array}{rrr|rrr}

1 & 1 & 2 & 1 &0 & 0 \\

0 & 1 & 1 & 1 & 0 & -1 \\

0 & 0 & 1 & 0 & 1 & 0 \\

\end{array} \right]\\[10pt]
&\xrightarrow{R_1-R_2}

\left[\begin{array}{rrr|rrr}

1 & 0 & 1 & 0 &0 & 1 \\

0 & 1 & 1 & 1 & 0 & -1 \\

0 & 0 & 1 & 0 & 1 & 0 \\

\end{array} \right]
\xrightarrow{\substack{R_1-R_3\\ R_2-R_3}}

\left[\begin{array}{rrr|rrr}

1 & 0 & 0 & 0 &-1 & 1 \\

0 & 1 & 0 & 1 & -1 & -1 \\

0 & 0 & 1 & 0 & 1 & 0 \\

\end{array} \right].

\end{align*}

The left $3\times 3$ matrix part became the identity matrix $I$, thus $A$ is invertible (since it is row equivalent to $I$), and the inverse matrix $A^{-1}$ is given by the right $3\times 3$ matrix. Thus we have

\[A^{-1}=\begin{bmatrix}

0 & -1 & 1 \\

1 &-1 &-1 \\

0 & 1 & 0

\end{bmatrix}.\]

### Double check

Once you obtained the inverse matrix $A^{-1}$, then you should check that whether $A^{-1}A$ is the identity matrix.

If not, it means that you did a computational mistake somewhere.

## Midterm 1 problems and solutions

This is one of the midterm exam 1 problems of linear algebra (Math 2568) at the Ohio State University.

The following list is the problems and solutions/proofs of midterm exam 1 of linear algebra at the Ohio State University in Spring 2017.

- Problem 1 and its solution: Possibilities for the solution set of a system of linear equations
- Problem 2 and its solution: The vector form of the general solution of a system
- Problem 3 and its solution: Matrix operations (transpose and inverse matrices)
- Problem 4 and its solution: Linear combination
- Problem 5 and its solution (The current page): Inverse matrix
- Problem 6 and its solution: Nonsingular matrix satisfying a relation
- Problem 7 and its solution: Solve a system by the inverse matrix
- Problem 8 and its solution:A proof problem about nonsingular matrix

## Related Question.

**Problem**.

Find the inverse matrices of the following matrices if they exist.

\[\text{(a)} \begin{bmatrix}

1 & 3 & -2 \\

2 &3 &0 \\

0 & 1 & 1

\end{bmatrix}, \quad \text{(b)} \begin{bmatrix}

1 & 0 & 2 \\

-1 &-3 &2 \\

3 & 6 & -2

\end{bmatrix}.\]

The solutions are given in the post↴

Find the Inverse Matrices if Matrices are Invertible by Elementary Row Operations

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[…] The solution is given in the post↴ Find the Inverse Matrix of a $3times 3$ Matrix if Exists […]