Chapter 10, Problem 11T

Only one of the following matrices has an inverse. Find the determinant of each matrix, and use the determinants to identify the one that has an inverse. Then find the inverse.

$\begin{array}{rr}\hfill A=\left[\begin{array}{ccc}1& 4& 1\\ 0& 2& 0\\ 1& 0& 1\end{array}\right]& \hfill B=\left[\begin{array}{rrr}\hfill 1& \hfill 4& \hfill 0\\ \hfill 0& \hfill 2& \hfill 0\\ \hfill -3& \hfill 0& \hfill 1\end{array}\right]\end{array}$

To determine

The determinant of the matrices and on the basis of determinant calculate inverse.

Answer to Problem 11T

The determinant of $A$ is $0$, determinant of $B$ is $2$ and the inverse of the matrix B is $\left[\begin{array}{ccc}1& -2& 0\\ 0& \frac{1}{2}& 0\\ 3& -6& 1\end{array}\right]$.

Explanation of Solution

Given:

The given matrices are,

$A=\left[\begin{array}{ccc}1& 4& 1\\ 0& 2& 0\\ 1& 0& 1\end{array}\right]$ and $B=\left[\begin{array}{ccc}1& 4& 0\\ 0& 2& 0\\ -3& 0& 1\end{array}\right]$

Calculation:

The determinant of matrix $A$ is,

Expand the matrix by row $2$.

$\begin{array}{c}\det \left(A\right)=\left|\begin{array}{ccc}1& 4& 1\\ 0& 2& 0\\ 1& 0& 1\end{array}\right|\\ =-0\left|\begin{array}{cc}4& 1\\ 0& 1\end{array}\right|+2\left|\begin{array}{cc}1& 1\\ 1& 1\end{array}\right|-0\left|\begin{array}{cc}1& 4\\ 0& 0\end{array}\right|\\ =0\end{array}$

The determinant of matrix A is 0 therefore the inverse of the matrix is not possible.

Find the determinant of matrix $B$.

Expand the matrix by row $2$.

$\begin{array}{c}\det \left(B\right)=\left|\begin{array}{ccc}1& 4& 0\\ 0& 2& 0\\ -3& 0& 1\end{array}\right|\\ =-0\left|\begin{array}{cc}4& 0\\ 0& 1\end{array}\right|+2\left|\begin{array}{cc}1& 0\\ -3& 1\end{array}\right|-0\left|\begin{array}{cc}1& 4\\ -3& 0\end{array}\right|\\ =2\end{array}$

The determinant of the matrix $B$ is non zero, so the inverse of this matrix is possible.

The given matrix is,

$B=\left[\begin{array}{ccc}1& 4& 0\\ 0& 2& 0\\ -3& 0& 1\end{array}\right]$

Write the matrix of $3\times 6$ order whose left half is $B$ and whose right half is the identity matrix.

$\left[\begin{array}{ccccccc}1& 4& 0& & 1& 0& 0\\ 0& 2& 0& & 0& 1& 0\\ -3& 0& 1& & 0& 0& 1\end{array}\right]$

Transform the left half portion of the above matrix into identity matrix by using elementary row operation.

Apply row operation $R_3\to R_3+R_1$ in above matrix.

$\left[\begin{array}{ccccccc}1& 4& 0& & 1& 0& 0\\ 0& 2& 0& & 0& 1& 0\\ 0& 12& 1& & 3& 0& 1\end{array}\right]$

Apply row operations $R_1\to R_1-2R_2$ and $R_3\to R_3-6R_2$ in above matrix.

$\left[\begin{array}{ccccccc}1& 0& 0& & 1& -2& 0\\ 0& 2& 0& & 0& 1& 0\\ 0& 0& 1& & 3& -6& 1\end{array}\right]$

Apply row operation $R_2\to \frac{1}{2}{R}_2$ in above matrix.

$\left[\begin{array}{ccccccc}1& 0& 0& & 1& -2& 0\\ 0& 1& 0& & 0& \frac{1}{2}& 0\\ 0& 0& 1& & 3& -6& 1\end{array}\right]$

The inverse of the matrix $B=\left[\begin{array}{ccc}1& 4& 0\\ 0& 2& 0\\ -3& 0& 1\end{array}\right]$ is $\left[\begin{array}{ccc}1& -2& 0\\ 0& \frac{1}{2}& 0\\ 3& -6& 1\end{array}\right]$.

Thus, the determinant of $A$ is $0$, determinant of $B$ is $2$ and the inverse of the matrix B is $\left[\begin{array}{ccc}1& -2& 0\\ 0& \frac{1}{2}& 0\\ 3& -6& 1\end{array}\right]$.