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.

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

Related Question.

The solutions are given in the post↴
Find the Inverse Matrices if Matrices are Invertible by Elementary Row Operations

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