A Few Matrix Applications

Example 1 (Gaussian elimination):

Solve:

$\begin{cases} x + y = 3 \\ 2x + 3y = 7 \end{cases}$

Augmented matrix:

$\begin{bmatrix} 1 & 1 & | & 3 \\ 2 & 3 & | & 7 \end{bmatrix}$