# Rank of Matrix/Examples/Arbitrary Matrix 5

## Examples of Rank of Matrix

Let $\mathbf A = \begin {bmatrix} -1 & 0 & 1 & 2 & 3 \\ 0 & 1 & 2 & 3 & 4 \\ -1 & -2 & -3 & -4 & -5 \end {bmatrix}$

The rank of $\mathbf A$ is $2$.

## Proof

From Matrix is Row Equivalent to Echelon Matrix: Arbitrary Matrix $5$, the echelon form $\mathbf E$ of $\mathbf A$ is:

$\mathbf E = \begin {bmatrix} 1 & 0 & -1 & -2 & -3 \\ 0 & 1 & 2 & 3 & 4 \\ 0 & 0 & 0 & 0 & 0 \end {bmatrix}$

There are $2$ non-zero rows in $\mathbf E$.

The result follows by definition of rank of matrix.

$\blacksquare$