Algebra lineare Esempi

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

Passaggio 1

Sottrai $m$ da entrambi i lati dell'equazione.

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

Passaggio 2

Sottrai $n$ da entrambi i lati dell'equazione.

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

Passaggio 3

Sottrai $o$ da entrambi i lati dell'equazione.

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

Passaggio 4

Sottrai $p$ da entrambi i lati dell'equazione.

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

Passaggio 5

Sottrai $q$ da entrambi i lati dell'equazione.

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

Passaggio 6

Sottrai $t$ da entrambi i lati dell'equazione.

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

Passaggio 7

Sottrai $u$ da entrambi i lati dell'equazione.

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

Passaggio 8

Scrivi il sistema di equazioni sotto forma di matrice.

Passaggio 9

Trova la forma ridotta a scala per righe di Echelon.

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Passaggio 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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Passaggio 9.1.1

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

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

Passaggio 9.1.2

Semplifica $R_{7}$ .

Passaggio 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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Passaggio 9.2.1

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

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

Passaggio 9.2.2

Semplifica $R_{7}$ .

Passaggio 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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Passaggio 9.3.1

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

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

Passaggio 9.3.2

Semplifica $R_{7}$ .

Passaggio 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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Passaggio 9.4.1

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

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

Passaggio 9.4.2

Semplifica $R_{7}$ .

Passaggio 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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Passaggio 9.5.1

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

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

Passaggio 9.5.2

Semplifica $R_{7}$ .

Passaggio 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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Passaggio 9.6.1

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

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

Passaggio 9.6.2

Semplifica $R_{7}$ .

Passaggio 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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Passaggio 9.7.1

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

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

Passaggio 9.7.2

Semplifica $R_{7}$ .

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

Passaggio 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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Passaggio 9.8.1

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

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

Passaggio 9.8.2

Semplifica $R_{6}$ .

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

Passaggio 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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Passaggio 9.9.1

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

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

Passaggio 9.9.2

Semplifica $R_{5}$ .

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

Passaggio 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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Passaggio 9.10.1

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

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

Passaggio 9.10.2

Semplifica $R_{4}$ .

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

Passaggio 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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Passaggio 9.11.1

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

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

Passaggio 9.11.2

Semplifica $R_{3}$ .

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

Passaggio 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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Passaggio 9.12.1

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

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

Passaggio 9.12.2

Semplifica $R_{2}$ .

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

Passaggio 9.13

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

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Passaggio 9.13.1

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

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

Passaggio 9.13.2

Semplifica $R_{1}$ .

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

Passaggio 10

Utilizza la matrice risultante per determinare le soluzioni finali del sistema di equazioni.

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

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

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

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

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

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

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

Passaggio 11

Sposta tutti i termini non contenenti $a$ sul lato destro dell'equazione.

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Passaggio 11.1

Somma $\frac{m}{2}$ a entrambi i lati dell'equazione.

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

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

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

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

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

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

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

Passaggio 11.2

Sottrai $\frac{n}{2}$ da entrambi i lati dell'equazione.

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

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

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

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

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

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

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

Passaggio 11.3

Somma $\frac{o}{2}$ a entrambi i lati dell'equazione.

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

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

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

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

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

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

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

Passaggio 11.4

Sottrai $\frac{p}{2}$ da entrambi i lati dell'equazione.

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

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

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

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

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

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

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

Passaggio 11.5

Somma $\frac{q}{2}$ a entrambi i lati dell'equazione.

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

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

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

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

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

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

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

Passaggio 11.6

Sottrai $\frac{t}{2}$ da entrambi i lati dell'equazione.

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

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

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

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

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

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

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

Passaggio 11.7

Somma $\frac{u}{2}$ a entrambi i lati dell'equazione.

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