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,

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