Rank of Matrix/Examples/Arbitrary Matrix 5
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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.
$\blacksquare$
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 $1.5 \ \text {(e)}$