Science:Math Exam Resources/Courses/MATH152/April 2016/Question A 23/Solution 2

Alternatively, we can also use row reduction.

Form the augmented matrix $[A|I]=$ $\left[\begin{array}{cccccc}2& 1& 3& 1& 0& 0\\ 1& 0& 1& 0& 1& 0\\ 0& 1& 2& 0& 0& 1\end{array}\right]$

Interchange $R_2$ and $R_1$ : $\left[\begin{array}{cccccc}1& 0& 1& 0& 1& 0\\ 2& 1& 3& 1& 0& 0\\ 0& 1& 2& 0& 0& 1\end{array}\right]$

$R_2-2R_1\to R_2$ : $\left[\begin{array}{cccccc}1& 0& 1& 0& 1& 0\\ 0& 1& 1& 1& -2& 0\\ 0& 1& 2& 0& 0& 1\end{array}\right]$

$R_3-R_2\to R_3$ : $\left[\begin{array}{cccccc}1& 0& 1& 0& 1& 0\\ 0& 1& 1& 1& -2& 0\\ 0& 0& 1& -1& 2& 1\end{array}\right]$

$R_1-R_3\to R_1$ : $\left[\begin{array}{cccccc}1& 0& 0& 1& -1& -1\\ 0& 1& 1& 1& -2& 0\\ 0& 0& 1& -1& 2& 1\end{array}\right]$

$R_2-R_3\to R_2$ : $\left[\begin{array}{cccccc}1& 0& 0& 1& -1& -1\\ 0& 1& 0& 2& -4& -1\\ 0& 0& 1& -1& 2& 1\end{array}\right]$

Therefore,

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