## Rank and Nullity of a Matrix

The Rank of a matrix is the number of non zero rows in the row reduced form of the matrx and the nullity is the number of zero rows in the row reduced form
Example:
$\mathbf{M} = \left( \begin{array}{ccc} 1 & 2 & 1 \\ 1 & -1 & 0 \\ -1 & -2 & -1 \\ 2 & 1 & 1 \end{array} \right)$

$\left( \begin{array}{ccc} 1 & 2 & 1 \\ 0 & -3 & -1 \\ 0 & 0 & 0 \\ 0 & 3 & -1 \end{array} \right) \begin{array}{c} \\ R2-R1 \\ R3+R1 \\R4-2*R1 \end{array}$

$\left( \begin{array}{ccc} 1 & 2 & 1 \\ 0 & -3 & -1 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{array} \right) \begin{array}{c} \\ \\ \\R4-R2 \end{array}$

There are two non zero rows so the rank of the matrix is 2 and there are two zero rows so the nullity is 2.