Naïve Gaussian elimination in matrix notation

Naïve Gaussian elimination can be implemented using matrix notation in the following way. Consider our system of peaches, apples and bananas.

$\left[\begin{array}{ccc}1& 4& 2\\ 2& 3& 1\\ 3& 2& 3\end{array}\right]\left[\begin{array}{c}p\\ a\\ b\end{array}\right]=\left[\begin{array}{c}3\\ 2.2\\ 3.2\end{array}\right]$

Rewrite this with only the numerical values as

$\left[\begin{array}{cccc}1& 4& 2& 3\\ 2& 3& 1& 2.2\\ 3& 2& 3& 3.2\end{array}\right]$

Subtract 2 times row one from row two. Replace row two with the new values.

$\left[\begin{array}{cccc}1& 4& 2& 3\\ 0& -5& -3& -3.8\\ 3& 2& 3& 3.2\end{array}\right]$

Subtract 3 times row one from row three. Replace row three with the new values.

$\left[\begin{array}{cccc}1& 4& 2& 3\\ 0& -5& -3& -3.8\\ 0& -10& -3& -5.8\end{array}\right]$

Subtract 2 times row two from row three and replace row three with the new values.

$\left[\begin{array}{cccc}1& 4& 2& 3\\ 0& -5& -3& -3.8\\ 0& 0& 3& 1.8\end{array}\right]$

This matrix is now in upper triangular form and we can easily back substitute to solve for b, then a and then p.