Matrix is Row Equivalent to Echelon Matrix/Examples/Arbitrary Matrix 1

Examples of Use of Matrix is Row Equivalent to Echelon Matrix

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

This can be converted into the echelon form:

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

Proof

Using Row Operation to Clear First Column of Matrix we obtain:

$\mathbf B = \begin {bmatrix} 1 & 1 & 0 \\ 0 & 1 & 0 \\ 0 & 1 & 1 \\ \end {bmatrix}$

which happens on the first step, exchanging row $1$ with row $3$.

Then we investigate the submatrix:

$\mathbf B' = \begin {bmatrix} 1 & 0 \\ 1 & 1 \\ \end {bmatrix}$

Using Row Operation to Clear First Column of Matrix we obtain:

$\mathbf C' = \begin {bmatrix} 1 & 0 \\ 0 & 1 \\ \end {bmatrix}$

which is obtained by adding $-1$ of row $1$ of $\mathbf B'$ to row $2$ of $\mathbf B'$.

This is the same as adding $-1$ of row $2$ of $\mathbf B$ to row $3$ of $\mathbf B$.

Thus we are left with:

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

$\blacksquare$

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 $1.4 \ \text {(a)}$