Matematika Berhingga Contoh

$[\begin{array}{c} 1 & 12 & 4 & 1 & 0 & 0 & 56 \\ 2 & 3 & 1 & 0 & 1 & 0 & 9 \\ - 1 & - 9 & - 2 & 0 & 0 & 1 & 0 \end{array}]$

Langkah 1

Tentukan bentuk eselon baris yang dikurangi.

Ketuk untuk lebih banyak langkah...

Langkah 1.1

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

Ketuk untuk lebih banyak langkah...

Langkah 1.1.1

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

$[\begin{array}{c} 1 & 12 & 4 & 1 & 0 & 0 & 56 \\ 2 - 2 \cdot 1 & 3 - 2 \cdot 12 & 1 - 2 \cdot 4 & 0 - 2 \cdot 1 & 1 - 2 \cdot 0 & 0 - 2 \cdot 0 & 9 - 2 \cdot 56 \\ - 1 & - 9 & - 2 & 0 & 0 & 1 & 0 \end{array}]$

Langkah 1.1.2

Sederhanakan $R_{2}$ .

$[\begin{array}{c} 1 & 12 & 4 & 1 & 0 & 0 & 56 \\ 0 & - 21 & - 7 & - 2 & 1 & 0 & - 103 \\ - 1 & - 9 & - 2 & 0 & 0 & 1 & 0 \end{array}]$

Langkah 1.2

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

Ketuk untuk lebih banyak langkah...

Langkah 1.2.1

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

$[\begin{array}{c} 1 & 12 & 4 & 1 & 0 & 0 & 56 \\ 0 & - 21 & - 7 & - 2 & 1 & 0 & - 103 \\ - 1 + 1 \cdot 1 & - 9 + 1 \cdot 12 & - 2 + 1 \cdot 4 & 0 + 1 \cdot 1 & 0 + 0 & 1 + 0 & 0 + 1 \cdot 56 \end{array}]$

Langkah 1.2.2

Sederhanakan $R_{3}$ .

$[\begin{array}{c} 1 & 12 & 4 & 1 & 0 & 0 & 56 \\ 0 & - 21 & - 7 & - 2 & 1 & 0 & - 103 \\ 0 & 3 & 2 & 1 & 0 & 1 & 56 \end{array}]$

Langkah 1.3

Multiply each element of $R_{2}$ by $- \frac{1}{21}$ to make the entry at $2, 2$ a $1$ .

Ketuk untuk lebih banyak langkah...

Langkah 1.3.1

Multiply each element of $R_{2}$ by $- \frac{1}{21}$ to make the entry at $2, 2$ a $1$ .

$[\begin{array}{c} 1 & 12 & 4 & 1 & 0 & 0 & 56 \\ - \frac{1}{21} \cdot 0 & - \frac{1}{21} \cdot - 21 & - \frac{1}{21} \cdot - 7 & - \frac{1}{21} \cdot - 2 & - \frac{1}{21} \cdot 1 & - \frac{1}{21} \cdot 0 & - \frac{1}{21} \cdot - 103 \\ 0 & 3 & 2 & 1 & 0 & 1 & 56 \end{array}]$

Langkah 1.3.2

Sederhanakan $R_{2}$ .

$[\begin{array}{c} 1 & 12 & 4 & 1 & 0 & 0 & 56 \\ 0 & 1 & \frac{1}{3} & \frac{2}{21} & - \frac{1}{21} & 0 & \frac{103}{21} \\ 0 & 3 & 2 & 1 & 0 & 1 & 56 \end{array}]$

Langkah 1.4

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

Ketuk untuk lebih banyak langkah...

Langkah 1.4.1

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

$[\begin{array}{c} 1 & 12 & 4 & 1 & 0 & 0 & 56 \\ 0 & 1 & \frac{1}{3} & \frac{2}{21} & - \frac{1}{21} & 0 & \frac{103}{21} \\ 0 - 3 \cdot 0 & 3 - 3 \cdot 1 & 2 - 3 (\frac{1}{3}) & 1 - 3 (\frac{2}{21}) & 0 - 3 (- \frac{1}{21}) & 1 - 3 \cdot 0 & 56 - 3 (\frac{103}{21}) \end{array}]$

Langkah 1.4.2

Sederhanakan $R_{3}$ .

$[\begin{array}{c} 1 & 12 & 4 & 1 & 0 & 0 & 56 \\ 0 & 1 & \frac{1}{3} & \frac{2}{21} & - \frac{1}{21} & 0 & \frac{103}{21} \\ 0 & 0 & 1 & \frac{5}{7} & \frac{1}{7} & 1 & \frac{289}{7} \end{array}]$

Langkah 1.5

Perform the row operation $R_{2} = R_{2} - \frac{1}{3} R_{3}$ to make the entry at $2, 3$ a $0$ .

Ketuk untuk lebih banyak langkah...

Langkah 1.5.1

Perform the row operation $R_{2} = R_{2} - \frac{1}{3} R_{3}$ to make the entry at $2, 3$ a $0$ .

$[\begin{array}{c} 1 & 12 & 4 & 1 & 0 & 0 & 56 \\ 0 - \frac{1}{3} \cdot 0 & 1 - \frac{1}{3} \cdot 0 & \frac{1}{3} - \frac{1}{3} \cdot 1 & \frac{2}{21} - \frac{1}{3} \cdot \frac{5}{7} & - \frac{1}{21} - \frac{1}{3} \cdot \frac{1}{7} & 0 - \frac{1}{3} \cdot 1 & \frac{103}{21} - \frac{1}{3} \cdot \frac{289}{7} \\ 0 & 0 & 1 & \frac{5}{7} & \frac{1}{7} & 1 & \frac{289}{7} \end{array}]$

Langkah 1.5.2

Sederhanakan $R_{2}$ .

$[\begin{array}{c} 1 & 12 & 4 & 1 & 0 & 0 & 56 \\ 0 & 1 & 0 & - \frac{1}{7} & - \frac{2}{21} & - \frac{1}{3} & - \frac{62}{7} \\ 0 & 0 & 1 & \frac{5}{7} & \frac{1}{7} & 1 & \frac{289}{7} \end{array}]$

Langkah 1.6

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

Ketuk untuk lebih banyak langkah...

Langkah 1.6.1

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

$[\begin{array}{c} 1 - 4 \cdot 0 & 12 - 4 \cdot 0 & 4 - 4 \cdot 1 & 1 - 4 (\frac{5}{7}) & 0 - 4 (\frac{1}{7}) & 0 - 4 \cdot 1 & 56 - 4 (\frac{289}{7}) \\ 0 & 1 & 0 & - \frac{1}{7} & - \frac{2}{21} & - \frac{1}{3} & - \frac{62}{7} \\ 0 & 0 & 1 & \frac{5}{7} & \frac{1}{7} & 1 & \frac{289}{7} \end{array}]$

Langkah 1.6.2

Sederhanakan $R_{1}$ .

$[\begin{array}{c} 1 & 12 & 0 & - \frac{13}{7} & - \frac{4}{7} & - 4 & - \frac{764}{7} \\ 0 & 1 & 0 & - \frac{1}{7} & - \frac{2}{21} & - \frac{1}{3} & - \frac{62}{7} \\ 0 & 0 & 1 & \frac{5}{7} & \frac{1}{7} & 1 & \frac{289}{7} \end{array}]$

Langkah 1.7

Perform the row operation $R_{1} = R_{1} - 12 R_{2}$ to make the entry at $1, 2$ a $0$ .

Ketuk untuk lebih banyak langkah...

Langkah 1.7.1

Perform the row operation $R_{1} = R_{1} - 12 R_{2}$ to make the entry at $1, 2$ a $0$ .

$[\begin{array}{c} 1 - 12 \cdot 0 & 12 - 12 \cdot 1 & 0 - 12 \cdot 0 & - \frac{13}{7} - 12 (- \frac{1}{7}) & - \frac{4}{7} - 12 (- \frac{2}{21}) & - 4 - 12 (- \frac{1}{3}) & - \frac{764}{7} - 12 (- \frac{62}{7}) \\ 0 & 1 & 0 & - \frac{1}{7} & - \frac{2}{21} & - \frac{1}{3} & - \frac{62}{7} \\ 0 & 0 & 1 & \frac{5}{7} & \frac{1}{7} & 1 & \frac{289}{7} \end{array}]$

Langkah 1.7.2

Sederhanakan $R_{1}$ .

$[\begin{array}{c} 1 & 0 & 0 & - \frac{1}{7} & \frac{4}{7} & 0 & - \frac{20}{7} \\ 0 & 1 & 0 & - \frac{1}{7} & - \frac{2}{21} & - \frac{1}{3} & - \frac{62}{7} \\ 0 & 0 & 1 & \frac{5}{7} & \frac{1}{7} & 1 & \frac{289}{7} \end{array}]$

Langkah 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 $a_{33}$

Pivot Columns: $1, 2,$ and $3$