Rank of Matrix/Examples/Arbitrary Matrix 4

Examples of Rank of Matrix

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

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

Proof

From Matrix is Row Equivalent to Echelon Matrix: Arbitrary Matrix $4$, the echelon form $\mathbf E$ of $\mathbf A$ is:

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

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

The result follows by definition of rank of matrix.

$\blacksquare$

Sources

• 1998: Richard Kaye and Robert Wilson: Linear Algebra ... (previous) ... (next): Part $\text I$: Matrices and vector spaces: $1$ Matrices: $1.5$ Row and column operations: Exercise $1.5 \ \text {(d)}$