Matematika Berhingga Contoh

$[\begin{array}{c} 2 & 9 & - 1 & 0 \\ 5 & - 7 & 6 & 1 \\ 7 & 2 & 5 & 1 \end{array}]$

Langkah 1

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

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Langkah 1.1

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

$[\begin{array}{c} \frac{2}{2} & \frac{9}{2} & \frac{- 1}{2} & \frac{0}{2} \\ 5 & - 7 & 6 & 1 \\ 7 & 2 & 5 & 1 \end{array}]$

Langkah 1.2

Sederhanakan $R_{1}$ .

$[\begin{array}{c} 1 & \frac{9}{2} & - \frac{1}{2} & 0 \\ 5 & - 7 & 6 & 1 \\ 7 & 2 & 5 & 1 \end{array}]$

Langkah 2

Perform the row operation $R_{2} = R_{2} - 5 R_{1}$ to make the entry at $2, 1$ a $0$ .

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Langkah 2.1

Perform the row operation $R_{2} = R_{2} - 5 R_{1}$ to make the entry at $2, 1$ a $0$ .

$[\begin{array}{c} 1 & \frac{9}{2} & - \frac{1}{2} & 0 \\ 5 - 5 \cdot 1 & - 7 - 5 (\frac{9}{2}) & 6 - 5 (- \frac{1}{2}) & 1 - 5 \cdot 0 \\ 7 & 2 & 5 & 1 \end{array}]$

Langkah 2.2

Sederhanakan $R_{2}$ .

$[\begin{array}{c} 1 & \frac{9}{2} & - \frac{1}{2} & 0 \\ 0 & - \frac{59}{2} & \frac{17}{2} & 1 \\ 7 & 2 & 5 & 1 \end{array}]$

Langkah 3

Perform the row operation $R_{3} = R_{3} - 7 R_{1}$ to make the entry at $3, 1$ a $0$ .

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Langkah 3.1

Perform the row operation $R_{3} = R_{3} - 7 R_{1}$ to make the entry at $3, 1$ a $0$ .

$[\begin{array}{c} 1 & \frac{9}{2} & - \frac{1}{2} & 0 \\ 0 & - \frac{59}{2} & \frac{17}{2} & 1 \\ 7 - 7 \cdot 1 & 2 - 7 (\frac{9}{2}) & 5 - 7 (- \frac{1}{2}) & 1 - 7 \cdot 0 \end{array}]$

Langkah 3.2

Sederhanakan $R_{3}$ .

$[\begin{array}{c} 1 & \frac{9}{2} & - \frac{1}{2} & 0 \\ 0 & - \frac{59}{2} & \frac{17}{2} & 1 \\ 0 & - \frac{59}{2} & \frac{17}{2} & 1 \end{array}]$

Langkah 4

Multiply each element of $R_{2}$ by $- \frac{2}{59}$ to make the entry at $2, 2$ a $1$ .

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Langkah 4.1

Multiply each element of $R_{2}$ by $- \frac{2}{59}$ to make the entry at $2, 2$ a $1$ .

$[\begin{array}{c} 1 & \frac{9}{2} & - \frac{1}{2} & 0 \\ - \frac{2}{59} \cdot 0 & - \frac{2}{59} (- \frac{59}{2}) & - \frac{2}{59} \cdot \frac{17}{2} & - \frac{2}{59} \cdot 1 \\ 0 & - \frac{59}{2} & \frac{17}{2} & 1 \end{array}]$

Langkah 4.2

Sederhanakan $R_{2}$ .

$[\begin{array}{c} 1 & \frac{9}{2} & - \frac{1}{2} & 0 \\ 0 & 1 & - \frac{17}{59} & - \frac{2}{59} \\ 0 & - \frac{59}{2} & \frac{17}{2} & 1 \end{array}]$

Langkah 5

Perform the row operation $R_{3} = R_{3} + \frac{59}{2} R_{2}$ to make the entry at $3, 2$ a $0$ .

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Langkah 5.1

Perform the row operation $R_{3} = R_{3} + \frac{59}{2} R_{2}$ to make the entry at $3, 2$ a $0$ .

$[\begin{array}{c} 1 & \frac{9}{2} & - \frac{1}{2} & 0 \\ 0 & 1 & - \frac{17}{59} & - \frac{2}{59} \\ 0 + \frac{59}{2} \cdot 0 & - \frac{59}{2} + \frac{59}{2} \cdot 1 & \frac{17}{2} + \frac{59}{2} (- \frac{17}{59}) & 1 + \frac{59}{2} (- \frac{2}{59}) \end{array}]$

Langkah 5.2

Sederhanakan $R_{3}$ .

$[\begin{array}{c} 1 & \frac{9}{2} & - \frac{1}{2} & 0 \\ 0 & 1 & - \frac{17}{59} & - \frac{2}{59} \\ 0 & 0 & 0 & 0 \end{array}]$

Langkah 6

Perform the row operation $R_{1} = R_{1} - \frac{9}{2} R_{2}$ to make the entry at $1, 2$ a $0$ .

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Langkah 6.1

Perform the row operation $R_{1} = R_{1} - \frac{9}{2} R_{2}$ to make the entry at $1, 2$ a $0$ .

$[\begin{array}{c} 1 - \frac{9}{2} \cdot 0 & \frac{9}{2} - \frac{9}{2} \cdot 1 & - \frac{1}{2} - \frac{9}{2} (- \frac{17}{59}) & 0 - \frac{9}{2} (- \frac{2}{59}) \\ 0 & 1 & - \frac{17}{59} & - \frac{2}{59} \\ 0 & 0 & 0 & 0 \end{array}]$

Langkah 6.2

Sederhanakan $R_{1}$ .

$[\begin{array}{c} 1 & 0 & \frac{47}{59} & \frac{9}{59} \\ 0 & 1 & - \frac{17}{59} & - \frac{2}{59} \\ 0 & 0 & 0 & 0 \end{array}]$