10 - Reduced Row Echelon Form (RREF)

Reduced Row Echelon Form of $A$

1. First non-zero entry of any non-zero row is $1$
2. Any column that contains a leading $1$, all other entries in the column are $0$
3. Leading $1$ of a non-zero row appears to the right of leading $1$s above it
4. Any zero rows appear below the non-zero rows

Applying EROs to transform a matrix into RRE form is called row-reduction / reduction

Also commonly referred to as Gauss-Jordan Elimination

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Solving Systems in RRE Form

Let $(A\mid \mathbf{b})$ be a matrix in RRE form which represents $A\mathbf{x}=\mathbf{b}$ of $m$ equations in $n$ variables

1. No solutions iff last non-zero row of $(A\mid \mathbf{b})$ is

$$\left(\begin{array}{cccc}0& 0& \cdots & 0\mid 1\end{array}\right)$$

2. Unique solution iff non-zero rows of $(A\mid \mathbf{b})$ form the identity matrix $I_n$ (requires $m\ge n$)

3. Infinitely many solutions

Proof - TODO