A Few Matrix Applications

Example 1 (Gaussian elimination):

Row operation: $R_2 \leftarrow R_2 - 2R_1$

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

Which corresponds to

$\begin{cases} x + y = 3 \\ 0 + y = 1 \end{cases}$

Back substitution gives $y = 1$, $x = 2$.