Matrix is Row Equivalent to Echelon Matrix/Examples/Arbitrary Matrix 2
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Examples of Use of Matrix is Row Equivalent to Echelon Matrix
Let $\mathbf A = \begin {bmatrix} 1 & 1 & 1 & 1 \\ 2 & 3 & 4 & 5 \\ 3 & 4 & 5 & 6 \\ \end {bmatrix}$
This can be converted into the echelon form:
- $\mathbf E = \begin {bmatrix} 1 & 1 & 1 & 1 \\ 0 & 1 & 2 & 3 \\ 0 & 0 & 0 & 0 \\ \end {bmatrix}$
Proof
Using Row Operation to Clear First Column of Matrix we obtain:
- $\mathbf B = \begin {bmatrix} 1 & 1 & 1 & 1 \\ 0 & 1 & 2 & 3 \\ 0 & 1 & 2 & 3 \\ \end {bmatrix}$
which is obtained by:
Then we investigate the submatrix:
- $\mathbf B' = \begin {bmatrix} 1 & 2 & 3 \\ 1 & 2 & 3 \\ \end {bmatrix}$
Using Row Operation to Clear First Column of Matrix we obtain:
- $\mathbf C' = \begin {bmatrix} 1 & 2 & 3 \\ 0 & 0 & 0 \\ \end {bmatrix}$
which is obtained by adding $-1$ of row $1$ of $\mathbf B'$ to row $2$ of $\mathbf B'$.
Thus we are left with:
- $\mathbf E = \begin {bmatrix} 1 & 1 & 1 & 1 \\ 0 & 1 & 2 & 3 \\ 0 & 0 & 0 & 0 \\ \end {bmatrix}$
$\blacksquare$
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 $1.4 \ \text {(b)}$