Aljabar Linear Contoh

$a + d = m$ , $d + g = n$ , $g + h = o$ , $h + j = p$ , $j + k = q$ , $k + l = t$ , $l + a = u$

Langkah 1

Kurangkan $m$ dari kedua sisi persamaan tersebut.

$a + d - m = 0, d + g = n, g + h = o, h + j = p, j + k = q, k + l = t, l + a = u$

Langkah 2

Kurangkan $n$ dari kedua sisi persamaan tersebut.

$a + d - m = 0, d + g - n = 0, g + h = o, h + j = p, j + k = q, k + l = t, l + a = u$

Langkah 3

Kurangkan $o$ dari kedua sisi persamaan tersebut.

$a + d - m = 0, d + g - n = 0, g + h - o = 0, h + j = p, j + k = q, k + l = t, l + a = u$

Langkah 4

Kurangkan $p$ dari kedua sisi persamaan tersebut.

$a + d - m = 0, d + g - n = 0, g + h - o = 0, h + j - p = 0, j + k = q, k + l = t, l + a = u$

Langkah 5

Kurangkan $q$ dari kedua sisi persamaan tersebut.

$a + d - m = 0, d + g - n = 0, g + h - o = 0, h + j - p = 0, j + k - q = 0, k + l = t, l + a = u$

Langkah 6

Kurangkan $t$ dari kedua sisi persamaan tersebut.

$a + d - m = 0, d + g - n = 0, g + h - o = 0, h + j - p = 0, j + k - q = 0, k + l - t = 0, l + a = u$

Langkah 7

Kurangkan $u$ dari kedua sisi persamaan tersebut.

$a + d - m = 0, d + g - n = 0, g + h - o = 0, h + j - p = 0, j + k - q = 0, k + l - t = 0, l + a - u = 0$

Langkah 8

Tulis sistem persamaan tersebut dalam bentuk matriks.

Langkah 9

Tentukan bentuk eselon baris yang dikurangi.

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Langkah 9.1

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

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Langkah 9.1.1

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

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

Langkah 9.1.2

Sederhanakan $R_{7}$ .

Langkah 9.2

Perform the row operation $R_{7} = R_{7} + R_{2}$ to make the entry at $7, 2$ a $0$ .

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Langkah 9.2.1

Perform the row operation $R_{7} = R_{7} + R_{2}$ to make the entry at $7, 2$ a $0$ .

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

Langkah 9.2.2

Sederhanakan $R_{7}$ .

Langkah 9.3

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

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Langkah 9.3.1

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

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

Langkah 9.3.2

Sederhanakan $R_{7}$ .

Langkah 9.4

Perform the row operation $R_{7} = R_{7} + R_{4}$ to make the entry at $7, 4$ a $0$ .

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Langkah 9.4.1

Perform the row operation $R_{7} = R_{7} + R_{4}$ to make the entry at $7, 4$ a $0$ .

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

Langkah 9.4.2

Sederhanakan $R_{7}$ .

Langkah 9.5

Perform the row operation $R_{7} = R_{7} - R_{5}$ to make the entry at $7, 5$ a $0$ .

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Langkah 9.5.1

Perform the row operation $R_{7} = R_{7} - R_{5}$ to make the entry at $7, 5$ a $0$ .

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

Langkah 9.5.2

Sederhanakan $R_{7}$ .

Langkah 9.6

Perform the row operation $R_{7} = R_{7} + R_{6}$ to make the entry at $7, 6$ a $0$ .

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Langkah 9.6.1

Perform the row operation $R_{7} = R_{7} + R_{6}$ to make the entry at $7, 6$ a $0$ .

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

Langkah 9.6.2

Sederhanakan $R_{7}$ .

Langkah 9.7

Multiply each element of $R_{7}$ by $\frac{1}{2}$ to make the entry at $7, 7$ a $1$ .

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Langkah 9.7.1

Multiply each element of $R_{7}$ by $\frac{1}{2}$ to make the entry at $7, 7$ a $1$ .

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

Langkah 9.7.2

Sederhanakan $R_{7}$ .

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

Langkah 9.8

Perform the row operation $R_{6} = R_{6} - R_{7}$ to make the entry at $6, 7$ a $0$ .

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Langkah 9.8.1

Perform the row operation $R_{6} = R_{6} - R_{7}$ to make the entry at $6, 7$ a $0$ .

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

Langkah 9.8.2

Sederhanakan $R_{6}$ .

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

Langkah 9.9

Perform the row operation $R_{5} = R_{5} - R_{6}$ to make the entry at $5, 6$ a $0$ .

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Langkah 9.9.1

Perform the row operation $R_{5} = R_{5} - R_{6}$ to make the entry at $5, 6$ a $0$ .

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

Langkah 9.9.2

Sederhanakan $R_{5}$ .

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

Langkah 9.10

Perform the row operation $R_{4} = R_{4} - R_{5}$ to make the entry at $4, 5$ a $0$ .

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Langkah 9.10.1

Perform the row operation $R_{4} = R_{4} - R_{5}$ to make the entry at $4, 5$ a $0$ .

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

Langkah 9.10.2

Sederhanakan $R_{4}$ .

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

Langkah 9.11

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

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Langkah 9.11.1

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

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

Langkah 9.11.2

Sederhanakan $R_{3}$ .

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

Langkah 9.12

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

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Langkah 9.12.1

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

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

Langkah 9.12.2

Sederhanakan $R_{2}$ .

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

Langkah 9.13

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

Ketuk untuk lebih banyak langkah...

Langkah 9.13.1

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

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

Langkah 9.13.2

Sederhanakan $R_{1}$ .

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

Langkah 10

Gunakan matriks hasil untuk menyatakan penyelesaian akhir untuk sistem persamaan tersebut.

$a - \frac{m}{2} + \frac{n}{2} - \frac{o}{2} + \frac{p}{2} - \frac{q}{2} + \frac{t}{2} - \frac{u}{2} = 0$

$d - \frac{m}{2} - \frac{n}{2} + \frac{o}{2} - \frac{p}{2} + \frac{q}{2} - \frac{t}{2} + \frac{u}{2} = 0$

$g + \frac{m}{2} - \frac{n}{2} - \frac{o}{2} + \frac{p}{2} - \frac{q}{2} + \frac{t}{2} - \frac{u}{2} = 0$

$h - \frac{m}{2} + \frac{n}{2} - \frac{o}{2} - \frac{p}{2} + \frac{q}{2} - \frac{t}{2} + \frac{u}{2} = 0$

$j + \frac{m}{2} - \frac{n}{2} + \frac{o}{2} - \frac{p}{2} - \frac{q}{2} + \frac{t}{2} - \frac{u}{2} = 0$

$k - \frac{m}{2} + \frac{n}{2} - \frac{o}{2} + \frac{p}{2} - \frac{q}{2} - \frac{t}{2} + \frac{u}{2} = 0$

$l + \frac{m}{2} - \frac{n}{2} + \frac{o}{2} - \frac{p}{2} + \frac{q}{2} - \frac{t}{2} - \frac{u}{2} = 0$

Langkah 11

Pindahkan semua suku yang tidak mengandung $a$ ke sisi kanan dari persamaan.

Ketuk untuk lebih banyak langkah...

Langkah 11.1

Tambahkan $\frac{m}{2}$ ke kedua sisi persamaan.

$a + \frac{n}{2} - \frac{o}{2} + \frac{p}{2} - \frac{q}{2} + \frac{t}{2} - \frac{u}{2} = \frac{m}{2}$

$d - \frac{m}{2} - \frac{n}{2} + \frac{o}{2} - \frac{p}{2} + \frac{q}{2} - \frac{t}{2} + \frac{u}{2} = 0$

$g + \frac{m}{2} - \frac{n}{2} - \frac{o}{2} + \frac{p}{2} - \frac{q}{2} + \frac{t}{2} - \frac{u}{2} = 0$

$h - \frac{m}{2} + \frac{n}{2} - \frac{o}{2} - \frac{p}{2} + \frac{q}{2} - \frac{t}{2} + \frac{u}{2} = 0$

$j + \frac{m}{2} - \frac{n}{2} + \frac{o}{2} - \frac{p}{2} - \frac{q}{2} + \frac{t}{2} - \frac{u}{2} = 0$

$k - \frac{m}{2} + \frac{n}{2} - \frac{o}{2} + \frac{p}{2} - \frac{q}{2} - \frac{t}{2} + \frac{u}{2} = 0$

$l + \frac{m}{2} - \frac{n}{2} + \frac{o}{2} - \frac{p}{2} + \frac{q}{2} - \frac{t}{2} - \frac{u}{2} = 0$

Langkah 11.2

Kurangkan $\frac{n}{2}$ dari kedua sisi persamaan tersebut.

$a - \frac{o}{2} + \frac{p}{2} - \frac{q}{2} + \frac{t}{2} - \frac{u}{2} = \frac{m}{2} - \frac{n}{2}$

$d - \frac{m}{2} - \frac{n}{2} + \frac{o}{2} - \frac{p}{2} + \frac{q}{2} - \frac{t}{2} + \frac{u}{2} = 0$

$g + \frac{m}{2} - \frac{n}{2} - \frac{o}{2} + \frac{p}{2} - \frac{q}{2} + \frac{t}{2} - \frac{u}{2} = 0$

$h - \frac{m}{2} + \frac{n}{2} - \frac{o}{2} - \frac{p}{2} + \frac{q}{2} - \frac{t}{2} + \frac{u}{2} = 0$

$j + \frac{m}{2} - \frac{n}{2} + \frac{o}{2} - \frac{p}{2} - \frac{q}{2} + \frac{t}{2} - \frac{u}{2} = 0$

$k - \frac{m}{2} + \frac{n}{2} - \frac{o}{2} + \frac{p}{2} - \frac{q}{2} - \frac{t}{2} + \frac{u}{2} = 0$

$l + \frac{m}{2} - \frac{n}{2} + \frac{o}{2} - \frac{p}{2} + \frac{q}{2} - \frac{t}{2} - \frac{u}{2} = 0$

Langkah 11.3

Tambahkan $\frac{o}{2}$ ke kedua sisi persamaan.

$a + \frac{p}{2} - \frac{q}{2} + \frac{t}{2} - \frac{u}{2} = \frac{m}{2} - \frac{n}{2} + \frac{o}{2}$

$d - \frac{m}{2} - \frac{n}{2} + \frac{o}{2} - \frac{p}{2} + \frac{q}{2} - \frac{t}{2} + \frac{u}{2} = 0$

$g + \frac{m}{2} - \frac{n}{2} - \frac{o}{2} + \frac{p}{2} - \frac{q}{2} + \frac{t}{2} - \frac{u}{2} = 0$

$h - \frac{m}{2} + \frac{n}{2} - \frac{o}{2} - \frac{p}{2} + \frac{q}{2} - \frac{t}{2} + \frac{u}{2} = 0$

$j + \frac{m}{2} - \frac{n}{2} + \frac{o}{2} - \frac{p}{2} - \frac{q}{2} + \frac{t}{2} - \frac{u}{2} = 0$

$k - \frac{m}{2} + \frac{n}{2} - \frac{o}{2} + \frac{p}{2} - \frac{q}{2} - \frac{t}{2} + \frac{u}{2} = 0$

$l + \frac{m}{2} - \frac{n}{2} + \frac{o}{2} - \frac{p}{2} + \frac{q}{2} - \frac{t}{2} - \frac{u}{2} = 0$

Langkah 11.4

Kurangkan $\frac{p}{2}$ dari kedua sisi persamaan tersebut.

$a - \frac{q}{2} + \frac{t}{2} - \frac{u}{2} = \frac{m}{2} - \frac{n}{2} + \frac{o}{2} - \frac{p}{2}$

$d - \frac{m}{2} - \frac{n}{2} + \frac{o}{2} - \frac{p}{2} + \frac{q}{2} - \frac{t}{2} + \frac{u}{2} = 0$

$g + \frac{m}{2} - \frac{n}{2} - \frac{o}{2} + \frac{p}{2} - \frac{q}{2} + \frac{t}{2} - \frac{u}{2} = 0$

$h - \frac{m}{2} + \frac{n}{2} - \frac{o}{2} - \frac{p}{2} + \frac{q}{2} - \frac{t}{2} + \frac{u}{2} = 0$

$j + \frac{m}{2} - \frac{n}{2} + \frac{o}{2} - \frac{p}{2} - \frac{q}{2} + \frac{t}{2} - \frac{u}{2} = 0$

$k - \frac{m}{2} + \frac{n}{2} - \frac{o}{2} + \frac{p}{2} - \frac{q}{2} - \frac{t}{2} + \frac{u}{2} = 0$

$l + \frac{m}{2} - \frac{n}{2} + \frac{o}{2} - \frac{p}{2} + \frac{q}{2} - \frac{t}{2} - \frac{u}{2} = 0$

Langkah 11.5

Tambahkan $\frac{q}{2}$ ke kedua sisi persamaan.

$a + \frac{t}{2} - \frac{u}{2} = \frac{m}{2} - \frac{n}{2} + \frac{o}{2} - \frac{p}{2} + \frac{q}{2}$

$d - \frac{m}{2} - \frac{n}{2} + \frac{o}{2} - \frac{p}{2} + \frac{q}{2} - \frac{t}{2} + \frac{u}{2} = 0$

$g + \frac{m}{2} - \frac{n}{2} - \frac{o}{2} + \frac{p}{2} - \frac{q}{2} + \frac{t}{2} - \frac{u}{2} = 0$

$h - \frac{m}{2} + \frac{n}{2} - \frac{o}{2} - \frac{p}{2} + \frac{q}{2} - \frac{t}{2} + \frac{u}{2} = 0$

$j + \frac{m}{2} - \frac{n}{2} + \frac{o}{2} - \frac{p}{2} - \frac{q}{2} + \frac{t}{2} - \frac{u}{2} = 0$

$k - \frac{m}{2} + \frac{n}{2} - \frac{o}{2} + \frac{p}{2} - \frac{q}{2} - \frac{t}{2} + \frac{u}{2} = 0$

$l + \frac{m}{2} - \frac{n}{2} + \frac{o}{2} - \frac{p}{2} + \frac{q}{2} - \frac{t}{2} - \frac{u}{2} = 0$

Langkah 11.6

Kurangkan $\frac{t}{2}$ dari kedua sisi persamaan tersebut.

$a - \frac{u}{2} = \frac{m}{2} - \frac{n}{2} + \frac{o}{2} - \frac{p}{2} + \frac{q}{2} - \frac{t}{2}$

$d - \frac{m}{2} - \frac{n}{2} + \frac{o}{2} - \frac{p}{2} + \frac{q}{2} - \frac{t}{2} + \frac{u}{2} = 0$

$g + \frac{m}{2} - \frac{n}{2} - \frac{o}{2} + \frac{p}{2} - \frac{q}{2} + \frac{t}{2} - \frac{u}{2} = 0$

$h - \frac{m}{2} + \frac{n}{2} - \frac{o}{2} - \frac{p}{2} + \frac{q}{2} - \frac{t}{2} + \frac{u}{2} = 0$

$j + \frac{m}{2} - \frac{n}{2} + \frac{o}{2} - \frac{p}{2} - \frac{q}{2} + \frac{t}{2} - \frac{u}{2} = 0$

$k - \frac{m}{2} + \frac{n}{2} - \frac{o}{2} + \frac{p}{2} - \frac{q}{2} - \frac{t}{2} + \frac{u}{2} = 0$

$l + \frac{m}{2} - \frac{n}{2} + \frac{o}{2} - \frac{p}{2} + \frac{q}{2} - \frac{t}{2} - \frac{u}{2} = 0$

Langkah 11.7

Tambahkan $\frac{u}{2}$ ke kedua sisi persamaan.

$a = \frac{m}{2} - \frac{n}{2} + \frac{o}{2} - \frac{p}{2} + \frac{q}{2} - \frac{t}{2} + \frac{u}{2}$