ringearV

2021-02-08

Use elementary matrices to write ? in row reduced echelon form. Use the elementary matrices to find the inverse of
$A=\left[\begin{array}{ccc}1& 0& -2\\ 0& 2& 1\\ 0& 0& 1\end{array}\right]$

### Answer & Explanation

krolaniaN

Step 1
Let $A=\left[\begin{array}{ccc}1& 0& -2\\ 0& 2& 1\\ 0& 0& 1\end{array}\right]$
Row reduced echelon form
${R}_{1}\to {R}_{1}+2{R}_{3}$
$\sim \left[\begin{array}{ccc}1& 0& 0\\ 0& 2& 1\\ 0& 0& 1\end{array}\right]$
${R}_{2}\to {R}_{2}-{R}_{3}$
$\sim \left[\begin{array}{ccc}1& 0& 0\\ 0& 2& 0\\ 0& 0& 1\end{array}\right]$
Gauss Jordon elimination :to find the matrix
$\left[A/I\right]=\left[\begin{array}{ccccccc}1& 0& -2& |& 1& 0& 0\\ 0& 2& 1& |& 0& 1& 0\\ 0& 0& 1& |& 0& 0& 1\end{array}\right]$
${R}_{2}\to \frac{{R}_{2}}{2}\sim \left[\begin{array}{ccccccc}1& 0& -2& |& 1& 0& 0\\ 0& 1& \frac{1}{2}& |& 0& \frac{1}{2}& 0\\ 0& 0& 1& |& 0& 0& 1\end{array}\right]$
${R}_{1}\to {R}_{1}+2{R}_{3}$
$\sim \left[\begin{array}{ccccccc}1& 0& 0& |& 1& 0& 2\\ 0& 1& 0& |& 0& \frac{1}{2}& 0\\ 0& 0& 1& |& 0& 0& 1\end{array}\right]$
${R}_{2}\to {R}_{2}-\frac{{R}_{3}}{2}$
$\sim \left[\begin{array}{ccccccc}1& 0& 0& |& 1& 0& 2\\ 0& 1& 0& |& 0& \frac{1}{2}& -\frac{1}{2}\\ 0& 0& 1& |& 0& 0& 1\end{array}\right]$
$\therefore {A}^{-1}=\left[\begin{array}{ccc}1& 0& 2\\ 0& \frac{1}{2}& -\frac{1}{2}\\ 0& 0& 1\end{array}\right]$
Varification $A\cdot {A}^{-1}=\left[\begin{array}{ccc}1& 0& -2\\ 0& 2& 1\\ 0& 0& 1\end{array}\right]\left[\begin{array}{ccc}1& 0& 2\\ 0& \frac{1}{2}& -\frac{1}{2}\\ 0& 0& 1\end{array}\right]=\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]$

Jeffrey Jordon