Matematika Berhingga Contoh

$[\begin{array}{c} 1 & - 1 & 9 & - 7.5 \\ 4 & 1 & 4 & - 6.5 \\ 4 & 6 & 1 & 2.5 \end{array}]$

Langkah 1

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

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Langkah 1.1

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

$[\begin{array}{c} 1 & - 1 & 9 & - 7.5 \\ 4 - 4 \cdot 1 & 1 - 4 \cdot - 1 & 4 - 4 \cdot 9 & - 6.5 - 4 \cdot - 7.5 \\ 4 & 6 & 1 & 2.5 \end{array}]$

Langkah 1.2

Sederhanakan $R_{2}$ .

$[\begin{array}{c} 1 & - 1 & 9 & - 7.5 \\ 0 & 5 & - 32 & 23.5 \\ 4 & 6 & 1 & 2.5 \end{array}]$

Langkah 2

Perform the row operation $R_{3} = R_{3} - 4 R_{1}$ to make the entry at $3, 1$ a $0$ .

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Langkah 2.1

Perform the row operation $R_{3} = R_{3} - 4 R_{1}$ to make the entry at $3, 1$ a $0$ .

$[\begin{array}{c} 1 & - 1 & 9 & - 7.5 \\ 0 & 5 & - 32 & 23.5 \\ 4 - 4 \cdot 1 & 6 - 4 \cdot - 1 & 1 - 4 \cdot 9 & 2.5 - 4 \cdot - 7.5 \end{array}]$

Langkah 2.2

Sederhanakan $R_{3}$ .

$[\begin{array}{c} 1 & - 1 & 9 & - 7.5 \\ 0 & 5 & - 32 & 23.5 \\ 0 & 10 & - 35 & 32.5 \end{array}]$

Langkah 3

Multiply each element of $R_{2}$ by $\frac{1}{5}$ to make the entry at $2, 2$ a $1$ .

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Langkah 3.1

Multiply each element of $R_{2}$ by $\frac{1}{5}$ to make the entry at $2, 2$ a $1$ .

$[\begin{array}{c} 1 & - 1 & 9 & - 7.5 \\ \frac{0}{5} & \frac{5}{5} & \frac{- 32}{5} & \frac{23.5}{5} \\ 0 & 10 & - 35 & 32.5 \end{array}]$

Langkah 3.2

Sederhanakan $R_{2}$ .

$[\begin{array}{c} 1 & - 1 & 9 & - 7.5 \\ 0 & 1 & - \frac{32}{5} & 4.7 \\ 0 & 10 & - 35 & 32.5 \end{array}]$

Langkah 4

Perform the row operation $R_{3} = R_{3} - 10 R_{2}$ to make the entry at $3, 2$ a $0$ .

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Langkah 4.1

Perform the row operation $R_{3} = R_{3} - 10 R_{2}$ to make the entry at $3, 2$ a $0$ .

$[\begin{array}{c} 1 & - 1 & 9 & - 7.5 \\ 0 & 1 & - \frac{32}{5} & 4.7 \\ 0 - 10 \cdot 0 & 10 - 10 \cdot 1 & - 35 - 10 (- \frac{32}{5}) & 32.5 - 10 \cdot 4.7 \end{array}]$

Langkah 4.2

Sederhanakan $R_{3}$ .

$[\begin{array}{c} 1 & - 1 & 9 & - 7.5 \\ 0 & 1 & - \frac{32}{5} & 4.7 \\ 0 & 0 & 29 & - 14.5 \end{array}]$

Langkah 5

Multiply each element of $R_{3}$ by $\frac{1}{29}$ to make the entry at $3, 3$ a $1$ .

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Langkah 5.1

Multiply each element of $R_{3}$ by $\frac{1}{29}$ to make the entry at $3, 3$ a $1$ .

$[\begin{array}{c} 1 & - 1 & 9 & - 7.5 \\ 0 & 1 & - \frac{32}{5} & 4.7 \\ \frac{0}{29} & \frac{0}{29} & \frac{29}{29} & \frac{- 14.5}{29} \end{array}]$

Langkah 5.2

Sederhanakan $R_{3}$ .

$[\begin{array}{c} 1 & - 1 & 9 & - 7.5 \\ 0 & 1 & - \frac{32}{5} & 4.7 \\ 0 & 0 & 1 & - 0.5 \end{array}]$

Langkah 6

Perform the row operation $R_{2} = R_{2} + \frac{32}{5} R_{3}$ to make the entry at $2, 3$ a $0$ .

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Langkah 6.1

Perform the row operation $R_{2} = R_{2} + \frac{32}{5} R_{3}$ to make the entry at $2, 3$ a $0$ .

$[\begin{array}{c} 1 & - 1 & 9 & - 7.5 \\ 0 + \frac{32}{5} \cdot 0 & 1 + \frac{32}{5} \cdot 0 & - \frac{32}{5} + \frac{32}{5} \cdot 1 & 4.7 + \frac{32}{5} \cdot - 0.5 \\ 0 & 0 & 1 & - 0.5 \end{array}]$

Langkah 6.2

Sederhanakan $R_{2}$ .

$[\begin{array}{c} 1 & - 1 & 9 & - 7.5 \\ 0 & 1 & 0 & 1.5 \\ 0 & 0 & 1 & - 0.5 \end{array}]$

Langkah 7

Perform the row operation $R_{1} = R_{1} - 9 R_{3}$ to make the entry at $1, 3$ a $0$ .

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Langkah 7.1

Perform the row operation $R_{1} = R_{1} - 9 R_{3}$ to make the entry at $1, 3$ a $0$ .

$[\begin{array}{c} 1 - 9 \cdot 0 & - 1 - 9 \cdot 0 & 9 - 9 \cdot 1 & - 7.5 - 9 \cdot - 0.5 \\ 0 & 1 & 0 & 1.5 \\ 0 & 0 & 1 & - 0.5 \end{array}]$

Langkah 7.2

Sederhanakan $R_{1}$ .

$[\begin{array}{c} 1 & - 1 & 0 & - 3 \\ 0 & 1 & 0 & 1.5 \\ 0 & 0 & 1 & - 0.5 \end{array}]$

Langkah 8

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

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Langkah 8.1

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

$[\begin{array}{c} 1 + 0 & - 1 + 1 \cdot 1 & 0 + 0 & - 3 + 1 \cdot 1.5 \\ 0 & 1 & 0 & 1.5 \\ 0 & 0 & 1 & - 0.5 \end{array}]$

Langkah 8.2

Sederhanakan $R_{1}$ .

$[\begin{array}{c} 1 & 0 & 0 & - 1.5 \\ 0 & 1 & 0 & 1.5 \\ 0 & 0 & 1 & - 0.5 \end{array}]$