Linear Algebra/Matrix Inverses/Finding the Inverse of a Matrix

The inverse of a matrix may be found using several different methods. The method that is guaranteed to work is by augmenting a n×n matrix with $I_n$, and solving to the RREF.

An example

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

$...$

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

The inverse of the matrix is the second augmented matrix. In this case,

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