Linear Algebra

Determine Whether the Following Matrix Invertible. If So Find Its Inverse Matrix.

Ohio State University exam problems and solutions in mathematics

Problem 76

Let A be the matrix
\[\begin{bmatrix}
1 & -1 & 0 \\
0 &1 &-1 \\
0 & 0 & 1
\end{bmatrix}.\] Is the matrix $A$ invertible? If not, then explain why it isn’t invertible. If so, then find the inverse.

(The Ohio State University Linear Algebra Exam)

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Hint.

Consider the augmented matrix $[A|I]$, where $I$ is the $3 \times 3$ identity matrix.

Solution.

Consider the augmented matrix $[A|I]$ with the $3\times 3$ identity matrix $I$.
Reduce $[A|I]$ using elementary row operations as follows.
\begin{align*}
[A|I]&=\left[\begin{array}{rrr|rrr}
1 & -1 & 0 & 1 &0 & 0 \\
0 & 1 & -1 & 0 & 1 & 0 \\
0 & 0 & 1 & 0 & 0 & 1 \\
\end{array} \right] \\
&\xrightarrow{R_2+R_3}
\left[\begin{array}{rrr|rrr}
1 & -1 & 0 & 1 &0 & 0 \\
0 & 1 & 0 & 0 & 1 & 1 \\
0 & 0 & 1 & 0 & 0 & 1 \\
\end{array} \right] \\
&\xrightarrow{R_1+R_2}
\left[\begin{array}{rrr|rrr}
1 & 0 & 0 & 1 &1 & 1 \\
0 & 1 & 0 & 0 & 1 & 1 \\
0 & 0 & 1 & 0 & 0 & 1 \\
\end{array} \right].
\end{align*}

This is in reduced row echelon form and the left $3 \times 3$ part is the identity matrix. Hence $A$ is invertible and the inverse matrix is
\[A^{-1}=\begin{bmatrix}
1 & 1 & 1 \\
0 &1 &1 \\
0 & 0 & 1
\end{bmatrix}.\]

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