Q. 38 - Precalculus Enhanced with Graphing Utilities

Question

In Problems $31-40$, each matrix is nonsingular. Find the inverse of each matrix.

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

Step-by-Step Solution

Answer

The inverse of the matrix $\left[\begin{array}{ccc}1& 0& 2\\ -1& 2& 3\\ 1& -1& 0\end{array}\right]$ is, $\left[\begin{array}{ccc}3& -2& -4\\ 3& -2& -5\\ -1& 1& 2\end{array}\right]$.

Step 1 . Given information

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

We have to find the inverse of the given matrix.

Step 2 . First find the augmented matrix A | I 3

$\left[A|I_3\right]\to \left[\begin{array}{cccccc}1& 0& 2& 1& 0& 0\\ -1& 2& 3& 0& 1& 0\\ 1& -1& 0& 0& 0& 1\end{array}\right]$

Now transform it into $\left[I_3|A^{-1}\right]$ using roe transformations.

$$\begin{gathered} \left[\begin{array}{cccccc}1& 0& 2& 1& 0& 0\\ -1& 2& 3& 0& 1& 0\\ 1& -1& 0& 0& 0& 1\end{array}\right]\to \left\{R_2\to R_2+R_1;R_3\to R_3-R_1\right\}\left[\begin{array}{cccccc}1& 0& 2& 1& 0& 0\\ 0& 2& 5& 1& 1& 0\\ 0& -1& -2& -1& 0& 1\end{array}\right] \\ \to \left\{R_2\to R_2+R_3\right\}\left[\begin{array}{cccccc}1& 0& 2& 1& 0& 0\\ 0& 1& 3& 0& 1& 1\\ 0& -1& -2& -1& 0& 1\end{array}\right] \\ \to \left\{R_3\to R_3+R_2\right\}\left[\begin{array}{cccccc}1& 0& 2& 1& 0& 0\\ 0& 1& 3& 0& 1& 1\\ 0& 0& 1& -1& 1& 2\end{array}\right] \\ \to \left\{R_1\to R_1-2R_3;R_2\to R_2-3R_3\right\}\left[\begin{array}{cccccc}1& 0& 0& 3& -2& -4\\ 0& 1& 0& 3& -2& -5\\ 0& 0& 1& -1& 1& 2\end{array}\right] \end{gathered}$$

Step 3 . The augmented matrix A | I 3 is now in the reduced echelon form and the identity matrix I 3 is on the left side of the vertical bar and A - 1 is on the right side of the vertical bar.

$A^{-1}=\left[\begin{array}{ccc}3& -2& -4\\ 3& -2& -5\\ -1& 1& 2\end{array}\right]$.