Algebra Examples

$[\begin{array}{c} 0 & 1 & 0 & 2 \\ 1 & 2 & - 1 & - 2 \end{array}]$

Assign the matrix the name $A$ to simplify the descriptions throughout the problem.

$A = [\begin{array}{c} 0 & 1 & 0 & 2 \\ 1 & 2 & - 1 & - 2 \end{array}]$

Find the reduced row echelon form of the matrix.

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Exchange row $2$ and row $1$ to organize the zeros into position.

$[\begin{array}{c} 1 & 2 & - 1 & - 2 \\ 0 & 1 & 0 & 2 \end{array}]$

$R_{2} \leftrightarrow R$

Perform the row operation $R_{1} = - 2 \cdot R_{2} + R_{1}$ on $R_{1}$ (row $1$ ) in order to convert some elements in the row to $0$ .

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Replace $R_{1}$ (row $1$ ) with the row operation $R_{1} = - 2 \cdot R_{2} + R_{1}$ in order to convert some elements in the row to the desired value $0$ .

$[\begin{array}{c} - 2 \cdot R_{2} + R_{1} & - 2 \cdot R_{2} + R_{1} & - 2 \cdot R_{2} + R_{1} & - 2 \cdot R_{2} + R_{1} \\ 0 & 1 & 0 & 2 \end{array}]$

$R_{1} = - 2 \cdot R_{2} + R_{1}$

Replace $R_{1}$ (row $1$ ) with the actual values of the elements for the row operation $R_{1} = - 2 \cdot R_{2} + R_{1}$ .

$[\begin{array}{c} (- 2) \cdot (0) + 1 & (- 2) \cdot (1) + 2 & (- 2) \cdot (0) - 1 & (- 2) \cdot (2) - 2 \\ 0 & 1 & 0 & 2 \end{array}]$

$R_{1} = - 2 \cdot R_{2} + R_{1}$

Simplify $R_{1}$ (row $1$ ).

$[\begin{array}{c} 1 & 0 & - 1 & - 6 \\ 0 & 1 & 0 & 2 \end{array}]$

$D$ represents the set of the pivot columns in the row-reduced form of the matrix.

$D = {1, 2}$

Rank is the dimension of the column space of $A$ .

$r (A) = d i m (D)$

The rank of $A$ is $2$ .

$2$

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