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

${\begin{bmatrix}1&3\\2&2\end{bmatrix}}{\Bigg |}{\begin{bmatrix}1&0\\0&1\end{bmatrix}}$

$...$

${\begin{bmatrix}1&0\\0&1\end{bmatrix}}{\Bigg |}{\begin{bmatrix}-{\cfrac {1}{2}}&{\cfrac {3}{4}}\\{\cfrac {1}{2}}&-{\cfrac {1}{4}}\end{bmatrix}}$

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

${\begin{bmatrix}-{\cfrac {1}{2}}&{\cfrac {3}{4}}\\{\cfrac {1}{2}}&-{\cfrac {1}{4}}\end{bmatrix}}$