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

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)

Contents

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
$A^{-1}=\begin{bmatrix} 1 & 1 & 1 \\ 0 &1 &1 \\ 0 & 0 & 1 \end{bmatrix}.$

Let $\Q$ denote the set of rational numbers (i.e., fractions of integers). Let $V$ denote the set of the form...