Algebra Examples

$[\begin{array}{c} 3 \\ 6 \end{array}]$

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

$A = [\begin{array}{c} 3 \\ 6 \end{array}]$

Find the reduced row echelon form of the matrix.

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Perform the row operation $R_{1} = \frac{1}{3} R_{1}$ 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 $R_{1} = \frac{1}{3} R_{1}$ in order to convert some elements in the row to the desired value $1$ .

$[\begin{array}{c} \frac{1}{3} R_{1} \\ 6 \end{array}]$

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

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

$[\begin{array}{c} (\frac{1}{3}) \cdot (3) \\ 6 \end{array}]$

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

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

$[\begin{array}{c} 1 \\ 6 \end{array}]$

Perform the row operation $R_{2} = - 6 \cdot R_{1} + R_{2}$ 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 $R_{2} = - 6 \cdot R_{1} + R_{2}$ in order to convert some elements in the row to the desired value $0$ .

$[\begin{array}{c} 1 \\ - 6 \cdot R_{1} + R_{2} \end{array}]$

$R_{2} = - 6 \cdot R_{1} + R_{2}$

Replace $R_{2}$ (row $2$ ) with the actual values of the elements for the row operation $R_{2} = - 6 \cdot R_{1} + R_{2}$ .

$[\begin{array}{c} 1 \\ (- 6) \cdot (1) + 6 \end{array}]$

$R_{2} = - 6 \cdot R_{1} + R_{2}$

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

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

$F$ represents the set of the free variable columns remaining in the row reduced matrix.

$F = {}$

Nullity is the dimension of the null space of $A$ , which is the same as the number of free variables columns remaining in the row reduced matrix.

$n (A) = d i m (F)$

The nullity of $A$ is $1$ .

$1$

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