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$[\begin{array}{c} 12 & 3 & 1 & 0 & 0 & 15 \\ 1 & 4 & 0 & 1 & 0 & 12 \\ - 8 & - 32 & 0 & 0 & 1 & 0 \end{array}]$

Bước 1

Tìm dạng ma trận hàng bậc thang rút gọn.

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Bước 1.1

Multiply each element of $R_{1}$ by $\frac{1}{12}$ to make the entry at $1, 1$ a $1$ .

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Bước 1.1.1

Multiply each element of $R_{1}$ by $\frac{1}{12}$ to make the entry at $1, 1$ a $1$ .

$[\begin{array}{c} \frac{12}{12} & \frac{3}{12} & \frac{1}{12} & \frac{0}{12} & \frac{0}{12} & \frac{15}{12} \\ 1 & 4 & 0 & 1 & 0 & 12 \\ - 8 & - 32 & 0 & 0 & 1 & 0 \end{array}]$

Bước 1.1.2

Rút gọn $R_{1}$ .

$[\begin{array}{c} 1 & \frac{1}{4} & \frac{1}{12} & 0 & 0 & \frac{5}{4} \\ 1 & 4 & 0 & 1 & 0 & 12 \\ - 8 & - 32 & 0 & 0 & 1 & 0 \end{array}]$

Bước 1.2

Perform the row operation $R_{2} = R_{2} - R_{1}$ to make the entry at $2, 1$ a $0$ .

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Bước 1.2.1

Perform the row operation $R_{2} = R_{2} - R_{1}$ to make the entry at $2, 1$ a $0$ .

$[\begin{array}{c} 1 & \frac{1}{4} & \frac{1}{12} & 0 & 0 & \frac{5}{4} \\ 1 - 1 & 4 - \frac{1}{4} & 0 - \frac{1}{12} & 1 - 0 & 0 - 0 & 12 - \frac{5}{4} \\ - 8 & - 32 & 0 & 0 & 1 & 0 \end{array}]$

Bước 1.2.2

Rút gọn $R_{2}$ .

$[\begin{array}{c} 1 & \frac{1}{4} & \frac{1}{12} & 0 & 0 & \frac{5}{4} \\ 0 & \frac{15}{4} & - \frac{1}{12} & 1 & 0 & \frac{43}{4} \\ - 8 & - 32 & 0 & 0 & 1 & 0 \end{array}]$

Bước 1.3

Perform the row operation $R_{3} = R_{3} + 8 R_{1}$ to make the entry at $3, 1$ a $0$ .

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Bước 1.3.1

Perform the row operation $R_{3} = R_{3} + 8 R_{1}$ to make the entry at $3, 1$ a $0$ .

$[\begin{array}{c} 1 & \frac{1}{4} & \frac{1}{12} & 0 & 0 & \frac{5}{4} \\ 0 & \frac{15}{4} & - \frac{1}{12} & 1 & 0 & \frac{43}{4} \\ - 8 + 8 \cdot 1 & - 32 + 8 (\frac{1}{4}) & 0 + 8 (\frac{1}{12}) & 0 + 8 \cdot 0 & 1 + 8 \cdot 0 & 0 + 8 (\frac{5}{4}) \end{array}]$

Bước 1.3.2

Rút gọn $R_{3}$ .

$[\begin{array}{c} 1 & \frac{1}{4} & \frac{1}{12} & 0 & 0 & \frac{5}{4} \\ 0 & \frac{15}{4} & - \frac{1}{12} & 1 & 0 & \frac{43}{4} \\ 0 & - 30 & \frac{2}{3} & 0 & 1 & 10 \end{array}]$

Bước 1.4

Multiply each element of $R_{2}$ by $\frac{4}{15}$ to make the entry at $2, 2$ a $1$ .

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Bước 1.4.1

Multiply each element of $R_{2}$ by $\frac{4}{15}$ to make the entry at $2, 2$ a $1$ .

$[\begin{array}{c} 1 & \frac{1}{4} & \frac{1}{12} & 0 & 0 & \frac{5}{4} \\ \frac{4}{15} \cdot 0 & \frac{4}{15} \cdot \frac{15}{4} & \frac{4}{15} (- \frac{1}{12}) & \frac{4}{15} \cdot 1 & \frac{4}{15} \cdot 0 & \frac{4}{15} \cdot \frac{43}{4} \\ 0 & - 30 & \frac{2}{3} & 0 & 1 & 10 \end{array}]$

Bước 1.4.2

Rút gọn $R_{2}$ .

$[\begin{array}{c} 1 & \frac{1}{4} & \frac{1}{12} & 0 & 0 & \frac{5}{4} \\ 0 & 1 & - \frac{1}{45} & \frac{4}{15} & 0 & \frac{43}{15} \\ 0 & - 30 & \frac{2}{3} & 0 & 1 & 10 \end{array}]$

Bước 1.5

Perform the row operation $R_{3} = R_{3} + 30 R_{2}$ to make the entry at $3, 2$ a $0$ .

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Bước 1.5.1

Perform the row operation $R_{3} = R_{3} + 30 R_{2}$ to make the entry at $3, 2$ a $0$ .

$[\begin{array}{c} 1 & \frac{1}{4} & \frac{1}{12} & 0 & 0 & \frac{5}{4} \\ 0 & 1 & - \frac{1}{45} & \frac{4}{15} & 0 & \frac{43}{15} \\ 0 + 30 \cdot 0 & - 30 + 30 \cdot 1 & \frac{2}{3} + 30 (- \frac{1}{45}) & 0 + 30 (\frac{4}{15}) & 1 + 30 \cdot 0 & 10 + 30 (\frac{43}{15}) \end{array}]$

Bước 1.5.2

Rút gọn $R_{3}$ .

$[\begin{array}{c} 1 & \frac{1}{4} & \frac{1}{12} & 0 & 0 & \frac{5}{4} \\ 0 & 1 & - \frac{1}{45} & \frac{4}{15} & 0 & \frac{43}{15} \\ 0 & 0 & 0 & 8 & 1 & 96 \end{array}]$

Bước 1.6

Multiply each element of $R_{3}$ by $\frac{1}{8}$ to make the entry at $3, 4$ a $1$ .

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Bước 1.6.1

Multiply each element of $R_{3}$ by $\frac{1}{8}$ to make the entry at $3, 4$ a $1$ .

$[\begin{array}{c} 1 & \frac{1}{4} & \frac{1}{12} & 0 & 0 & \frac{5}{4} \\ 0 & 1 & - \frac{1}{45} & \frac{4}{15} & 0 & \frac{43}{15} \\ \frac{0}{8} & \frac{0}{8} & \frac{0}{8} & \frac{8}{8} & \frac{1}{8} & \frac{96}{8} \end{array}]$

Bước 1.6.2

Rút gọn $R_{3}$ .

$[\begin{array}{c} 1 & \frac{1}{4} & \frac{1}{12} & 0 & 0 & \frac{5}{4} \\ 0 & 1 & - \frac{1}{45} & \frac{4}{15} & 0 & \frac{43}{15} \\ 0 & 0 & 0 & 1 & \frac{1}{8} & 12 \end{array}]$

Bước 1.7

Perform the row operation $R_{2} = R_{2} - \frac{4}{15} R_{3}$ to make the entry at $2, 4$ a $0$ .

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Bước 1.7.1

Perform the row operation $R_{2} = R_{2} - \frac{4}{15} R_{3}$ to make the entry at $2, 4$ a $0$ .

$[\begin{array}{c} 1 & \frac{1}{4} & \frac{1}{12} & 0 & 0 & \frac{5}{4} \\ 0 - \frac{4}{15} \cdot 0 & 1 - \frac{4}{15} \cdot 0 & - \frac{1}{45} - \frac{4}{15} \cdot 0 & \frac{4}{15} - \frac{4}{15} \cdot 1 & 0 - \frac{4}{15} \cdot \frac{1}{8} & \frac{43}{15} - \frac{4}{15} \cdot 12 \\ 0 & 0 & 0 & 1 & \frac{1}{8} & 12 \end{array}]$

Bước 1.7.2

Rút gọn $R_{2}$ .

$[\begin{array}{c} 1 & \frac{1}{4} & \frac{1}{12} & 0 & 0 & \frac{5}{4} \\ 0 & 1 & - \frac{1}{45} & 0 & - \frac{1}{30} & - \frac{1}{3} \\ 0 & 0 & 0 & 1 & \frac{1}{8} & 12 \end{array}]$

Bước 1.8

Perform the row operation $R_{1} = R_{1} - \frac{1}{4} R_{2}$ to make the entry at $1, 2$ a $0$ .

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Bước 1.8.1

Perform the row operation $R_{1} = R_{1} - \frac{1}{4} R_{2}$ to make the entry at $1, 2$ a $0$ .

$[\begin{array}{c} 1 - \frac{1}{4} \cdot 0 & \frac{1}{4} - \frac{1}{4} \cdot 1 & \frac{1}{12} - \frac{1}{4} (- \frac{1}{45}) & 0 - \frac{1}{4} \cdot 0 & 0 - \frac{1}{4} (- \frac{1}{30}) & \frac{5}{4} - \frac{1}{4} (- \frac{1}{3}) \\ 0 & 1 & - \frac{1}{45} & 0 & - \frac{1}{30} & - \frac{1}{3} \\ 0 & 0 & 0 & 1 & \frac{1}{8} & 12 \end{array}]$

Bước 1.8.2

Rút gọn $R_{1}$ .

$[\begin{array}{c} 1 & 0 & \frac{4}{45} & 0 & \frac{1}{120} & \frac{4}{3} \\ 0 & 1 & - \frac{1}{45} & 0 & - \frac{1}{30} & - \frac{1}{3} \\ 0 & 0 & 0 & 1 & \frac{1}{8} & 12 \end{array}]$

Bước 2

The pivot positions are the locations with the leading $1$ in each row. The pivot columns are the columns that have a pivot position.

Pivot Positions: $a_{11}, a_{22},$ and $a_{34}$

Pivot Columns: $1, 2,$ and $4$