Question

Using elementary row transformations, find the inverse of the matrix

$\left[\begin{array}{ccc}1& 2& 3\\ 2& 5& 7\\ -2& -4& 5\end{array}\right]$

Answer 1

$\left[\begin{array}{ccc}1& 2& 3\\ 2& 5& 7\\ -2& -4& 5\end{array}\right]$

$\left[\begin{array}{ccccccc}1& 2& 3& |& 1& 0& 0\\ 2& 5& 7& |& 0& 1& 0\\ -2& -4& 5& |& 0& 0& 1\end{array}\right]$

eliminate first column

$\left[\begin{array}{ccccccc}1& 2& 3& |& 1& 0& 0\\ 0& 1& 1& |& -2& 1& 0\\ 0& 0& 11& |& 2& 0& 1\end{array}\right]$

eliminate second column

$\left[\begin{array}{ccccccc}1& 0& 1& |& 5& -2& 0\\ 0& 1& 1& |& -2& 1& 0\\ 0& 0& 11& |& 2& 0& 1\end{array}\right]$

divide third row by 11

$\left[\begin{array}{ccccccc}1& 0& 1& |& 5& -2& 0\\ 0& 1& 1& |& -2& 1& 0\\ 0& 0& 1& |& \frac{2}{11}& 0& \frac{1}{11}\end{array}\right]$

eliminate third row

$\left[\begin{array}{ccccccc}1& 0& 0& |& \frac{53}{11}& -2& \frac{-1}{11}\\ 0& 1& 0& |& \frac{-24}{11}& 1& \frac{-1}{11}\\ 0& 0& 1& |& \frac{2}{11}& 0& \frac{1}{11}\end{array}\right]$

inverse of matrix

$\left[\begin{array}{ccc}\frac{53}{11}& -2& \frac{-1}{11}\\ \frac{-24}{11}& 1& \frac{-1}{11}\\ \frac{2}{11}& 0& \frac{1}{11}\end{array}\right]$