Algebra Examples

$[\begin{array}{c} 2 & - 2 & 9 \\ 1 & 2 & 3 \end{array}]$

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

$A = [\begin{array}{c} 2 & - 2 & 9 \\ 1 & 2 & 3 \end{array}]$

Find the reduced row echelon form of the matrix.

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

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

$[\begin{array}{c} \frac{1}{2} R_{1} & \frac{1}{2} R_{1} & \frac{1}{2} R_{1} \\ 1 & 2 & 3 \end{array}]$

$R_{1} = \frac{1}{2} R_{1}$

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

$[\begin{array}{c} (\frac{1}{2}) \cdot (2) & (\frac{1}{2}) \cdot (- 2) & (\frac{1}{2}) \cdot (9) \\ 1 & 2 & 3 \end{array}]$

$R_{1} = \frac{1}{2} R_{1}$

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

$[\begin{array}{c} 1 & - 1 & \frac{9}{2} \\ 1 & 2 & 3 \end{array}]$

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

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

$[\begin{array}{c} 1 & - 1 & \frac{9}{2} \\ - 1 \cdot R_{1} + R_{2} & - 1 \cdot R_{1} + R_{2} & - 1 \cdot R_{1} + R_{2} \end{array}]$

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

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

$[\begin{array}{c} 1 & - 1 & \frac{9}{2} \\ (- 1) \cdot (1) + 1 & (- 1) \cdot (- 1) + 2 & (- 1) \cdot (\frac{9}{2}) + 3 \end{array}]$

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

Simplify $R_{2}$ (row $2$ ).

$[\begin{array}{c} 1 & - 1 & \frac{9}{2} \\ 0 & 3 & - \frac{3}{2} \end{array}]$

Perform the row operation $R_{2} = \frac{1}{3} R_{2}$ on $R_{2}$ (row $2$ ) in order to convert some elements in the row to $1$ .

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Replace $R_{2}$ (row $2$ ) with the row operation $R_{2} = \frac{1}{3} R_{2}$ in order to convert some elements in the row to the desired value $1$ .

$[\begin{array}{c} 1 & - 1 & \frac{9}{2} \\ \frac{1}{3} R_{2} & \frac{1}{3} R_{2} & \frac{1}{3} R_{2} \end{array}]$

$R_{2} = \frac{1}{3} R_{2}$

Replace $R_{2}$ (row $2$ ) with the actual values of the elements for the row operation $R_{2} = \frac{1}{3} R_{2}$ .

$[\begin{array}{c} 1 & - 1 & \frac{9}{2} \\ (\frac{1}{3}) \cdot (0) & (\frac{1}{3}) \cdot (3) & (\frac{1}{3}) \cdot (- \frac{3}{2}) \end{array}]$

$R_{2} = \frac{1}{3} R_{2}$

Simplify $R_{2}$ (row $2$ ).

$[\begin{array}{c} 1 & - 1 & \frac{9}{2} \\ 0 & 1 & - \frac{1}{2} \end{array}]$

Perform the row operation $R_{1} = R_{2} + R_{1}$ 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 $R_{1} = R_{2} + R_{1}$ in order to convert some elements in the row to the desired value $0$ .

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

$R_{1} = R_{2} + R_{1}$

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

$[\begin{array}{c} 0 + 1 & 1 - 1 & - \frac{1}{2} + \frac{9}{2} \\ 0 & 1 & - \frac{1}{2} \end{array}]$

$R_{1} = R_{2} + R_{1}$

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

$[\begin{array}{c} 1 & 0 & 4 \\ 0 & 1 & - \frac{1}{2} \end{array}]$

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

$D = (S E T, 1, 2)$

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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