Gaussian Elimination: a systematic way of solving the system

not always possible, when impossible, get Row Echelon Form

always reaches echolon form

If the number of equations equals the number of variables, the system in non-singular

Pivot: leading coefficients

(Row) Echelon Form (ref):

Rows at the form (0, ..., 0 | x) are at the bottom if exists

any pivot cutting is in a column to the left from any pivot below it // QUESTION: WHAT? Explain? pivot of a nonzero row is always strictly to the right of the leading coefficient of the row above it.

If the system is non-singular: Echolon Form is Triangular

there is exactly one solution

every column has a pivot entry

If the system is singular: at least one (0, ..., 0 | x) (but the opposite is not true)

if x == 0: infinite many solution if it is in n by n system

if x != 0: no solution in any system

Reduced Row Echelon Form (rref): if furthermore all of the leading coefficients are equal to 1

Canonical Form: can be viewed as solution

Triangular System: special Echelon Form which indicate exactly one solution

\begin{pmatrix}
\begin{array}{ccccc|c}
\circ & x & x & x & x & x\\
0 & \circ & x & x & x & x\\
0 & 0 & \circ & x & x & x\\
0 & 0 & 0 & \circ & x & x\\
0 & 0 & 0 & 0 & \circ & x\\
\end{array}
\end{pmatrix}

where \circ represent non-zero, and x can be zero or non-zero

Solving a system with infinite many solutions

If a system has infinite many solution, pick the variable that is not in Row-reduced form, then subsitute that with variable constant P, solve for P.