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

Problem 552

For each of the following $3\times 3$ matrices $A$, determine whether $A$ is invertible and find the inverse $A^{-1}$ if exists by computing the augmented matrix $[A|I]$, where $I$ is the $3\times 3$ identity matrix.

(a) $A=\begin{bmatrix} 1 & 3 & -2 \\ 2 &3 &0 \\ 0 & 1 & 1 \end{bmatrix}$

(b) $A=\begin{bmatrix} 1 & 0 & 2 \\ -1 &-3 &2 \\ 3 & 6 & -2 \end{bmatrix}$.

Elementary Row Operations and Inverse Matrices

Recall the following procedure of testing the invertibility of $A$ as well as finding the inverse matrix if exists.
If the augmented matrix $[A|I]$ is transformed into a matrix of the form $[I|B]$, then the matrix $A$ is invertible and the inverse matrix $A^{-1}$ is given by $B$.
If the reduced row echelon form matrix for $[A|I]$ is not of the form $[I|B]$, then the matrix $A$ is not invertible.

Solution.

(a) $A=\begin{bmatrix} 1 & 3 & -2 \\ 2 &3 &0 \\ 0 & 1 & 1 \end{bmatrix}$

We apply the elementary row operations as follows.
We have
\begin{align*}
&[A|I]= \left[\begin{array}{rrr|rrr}
1 & 3 & -2 & 1 &0 & 0 \\
2 & 3 & 0 & 0 & 1 & 0 \\
0 & 1 & -1 & 0 & 0 & 1 \\
\end{array} \right] \xrightarrow{R_2-2R_1}
\left[\begin{array}{rrr|rrr}
1 & 3 & -2 & 1 &0 & 0 \\
0 & -3 & 4 & -2 & 1 & 0 \\
0 & 1 & -1 & 0 & 0 & 1 \\
\end{array} \right]\6pt] &\xrightarrow{R_2\leftrightarrow R_3} \left[\begin{array}{rrr|rrr} 1 & 3 & -2 & 1 &0 & 0 \\ 0 & 1 & -1 & 0 & 0 & 1 \\ 0 & -3 & 4 & -2 & 1 & 0 \\ \end{array} \right] \xrightarrow{\substack{R_1-3R_2\\ R_3+3R_2}} \left[\begin{array}{rrr|rrr} 1 & 0& 1 & 1 &0 & -3 \\ 0 & 1 & -1 & 0 & 0 & 1 \\ 0 & 0 & 1 & -2 & 1 & 3 \\ \end{array} \right]\\[6pt] &\xrightarrow{\substack{R_1-R_3\\ R_2+R_3}} \left[\begin{array}{rrr|rrr} 1 & 0& 0 & 3 & -1 & -6 \\ 0 & 1 & 0 & -2 & 1 & 4 \\ 0 & 0 & 1 & -2 & 1 & 3 \\ \end{array} \right]. \end{align*} The left 3\times 3 part of the last matrix is the identity matrix. This implies that A is invertible and the inverse matrix is given by the right 3\times 3 matrix. Hence \[A^{-1}=\begin{bmatrix} 3 & -1 & -6 \\ -2 &1 &4 \\ -2 & 1 & 3 \end{bmatrix}.

(b) $A=\begin{bmatrix} 1 & 0 & 2 \\ -1 &-3 &2 \\ 3 & 6 & -2 \end{bmatrix}$.

Now we consider the matrix $A=\begin{bmatrix} 1 & 0 & 2 \\ -1 &-3 &2 \\ 3 & 6 & -2 \end{bmatrix}$.

Applying elementary row operations, we obtain
\begin{align*}
&[A|I]= \left[\begin{array}{rrr|rrr}
1 & 0 & 2 & 1 &0 & 0 \\
-1 & -3 & 2 & 0 & 1 & 0 \\
3 & 6 & -2 & 0 & 0 & 1 \\
\end{array} \right] \xrightarrow{\substack{R_2+R_1\\ R_3-3R_1}}
\left[\begin{array}{rrr|rrr}
1 & 0 & 2 & 1 &0 & 0 \\
0 & -3 & 4 & 1 & 1 & 0 \\
0 & 6 & -8 & -3 & 0 & 1 \\
\end{array} \right]\6pt] &\xrightarrow{R_3-2R_1} \left[\begin{array}{rrr|rrr} 1 & 0 & 2 & 1 &0 & 0 \\ 0 & -3 & 4 & 1 & 1 & 0 \\ 0 & 0 & 0 & -5 & -2 & 1 \\ \end{array} \right] \xrightarrow{\frac{-1}{3}R_2} \left[\begin{array}{rrr|rrr} 1 & 0 & 2 & 1 &0 & 0 \\ 0 & 1 & -4/3 & -1/3 & -1/3 & 0 \\ 0 & 0 & 0 & -5 & -2 & 1 \\ \end{array} \right]. \end{align*} The last matrix is in reduced row echelon form but the left 3\times 3 part is not the identity matrix I. It follows that the matrix A is not invertible. Related Question. Problem. 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.

This is a linear algebra exam problem at the Ohio State University.

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

More from my site

• Quiz 4: Inverse Matrix/ Nonsingular Matrix Satisfying a Relation (a) Find the inverse matrix of $A=\begin{bmatrix} 1 & 0 & 1 \\ 1 &0 &0 \\ 2 & 1 & 1 \end{bmatrix}$ if it exists. If you think there is no inverse matrix of $A$, then give a reason. (b) Find a nonsingular $2\times 2$ matrix $A$ such that $A^3=A^2B-3A^2,$ where […]
• Determine Whether the Following Matrix Invertible. If So Find Its Inverse Matrix. 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 […]
• The Inverse Matrix of an Upper Triangular Matrix with Variables Let $A$ be the following $3\times 3$ upper triangular matrix. $A=\begin{bmatrix} 1 & x & y \\ 0 &1 &z \\ 0 & 0 & 1 \end{bmatrix},$ where $x, y, z$ are some real numbers. Determine whether the matrix $A$ is invertible or not. If it is invertible, then find […]
• Find Values of $a$ so that the Matrix is Nonsingular Let $A$ be the following $3 \times 3$ matrix. $A=\begin{bmatrix} 1 & 1 & -1 \\ 0 &1 &2 \\ 1 & 1 & a \end{bmatrix}.$ Determine the values of $a$ so that the matrix $A$ is nonsingular.   Solution. We use the fact that a matrix is nonsingular if and only if […]
• For Which Choices of $x$ is the Given Matrix Invertible? Determine the values of $x$ so that the matrix $A=\begin{bmatrix} 1 & 1 & x \\ 1 &x &x \\ x & x & x \end{bmatrix}$ is invertible. For those values of $x$, find the inverse matrix $A^{-1}$.   Solution. We use the fact that a matrix is invertible […]
• Find the Inverse Matrix of a $3\times 3$ Matrix if Exists 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 […]
• Find Values of $h$ so that the Given Vectors are Linearly Independent Find the value(s) of $h$ for which the following set of vectors $\left \{ \mathbf{v}_1=\begin{bmatrix} 1 \\ 0 \\ 0 \end{bmatrix}, \mathbf{v}_2=\begin{bmatrix} h \\ 1 \\ -h \end{bmatrix}, \mathbf{v}_3=\begin{bmatrix} 1 \\ 2h \\ 3h+1 […] • Find a Nonsingular Matrix Satisfying Some Relation Determine whether there exists a nonsingular matrix A if \[A^2=AB+2A,$ where $B$ is the following matrix. If such a nonsingular matrix $A$ exists, find the inverse matrix $A^{-1}$. (a) \[B=\begin{bmatrix} -1 & 1 & -1 \\ 0 &-1 &0 \\ 1 & 2 & […]

2 Responses

1. 08/31/2017

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

2. 08/31/2017

[…] 3 & -1 & -6 \ -2 &1 &4 \ -2 & 1 & 3 end{bmatrix}.] (See the post Find the Inverse Matrices if Matrices are Invertible by Elementary Row Operations for details of how to find the inverse matrix of this […]

The Sum of Cosine Squared in an Inner Product Space

Let $\mathbf{v}$ be a vector in an inner product space $V$ over $\R$. Suppose that $\{\mathbf{u}_1, \dots, \mathbf{u}_n\}$ is an...

Close