الرياضيات المتناهية الأمثلة

$[\begin{array}{c} 1 & 1 & 0 & 0 & 1500 \\ 0 & 0 & 1 & 1 & 1500 \\ 1 & 0 & 1 & 0 & 1000 \\ 0 & 1 & 0 & 1 & 2000 \\ 1 & 4 & 1 & 2 & 8620 \end{array}]$

خطوة 1

Choose the row or column with the most $0$ elements. If there are no $0$ elements choose any row or column. Multiply every element in row $1$ by its cofactor and add.

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خطوة 1.1

Consider the corresponding sign chart.

$| \begin{array}{c} + & - & + & - & + \\ - & + & - & + & - \\ + & - & + & - & + \\ - & + & - & + & - \\ + & - & + & - & + \end{array} |$

خطوة 1.2

The cofactor is the minor with the sign changed if the indices match a $-$ position on the sign chart.

خطوة 1.3

The minor for $a_{11}$ is the determinant with row $1$ and column $1$ deleted.

$| \begin{array}{c} 0 & 1 & 1 & 1500 \\ 0 & 1 & 0 & 1000 \\ 1 & 0 & 1 & 2000 \\ 4 & 1 & 2 & 8620 \end{array} |$

خطوة 1.4

Multiply element $a_{11}$ by its cofactor.

$1 | \begin{array}{c} 0 & 1 & 1 & 1500 \\ 0 & 1 & 0 & 1000 \\ 1 & 0 & 1 & 2000 \\ 4 & 1 & 2 & 8620 \end{array} |$

خطوة 1.5

The minor for $a_{12}$ is the determinant with row $1$ and column $2$ deleted.

$| \begin{array}{c} 0 & 1 & 1 & 1500 \\ 1 & 1 & 0 & 1000 \\ 0 & 0 & 1 & 2000 \\ 1 & 1 & 2 & 8620 \end{array} |$

خطوة 1.6

Multiply element $a_{12}$ by its cofactor.

$- 1 | \begin{array}{c} 0 & 1 & 1 & 1500 \\ 1 & 1 & 0 & 1000 \\ 0 & 0 & 1 & 2000 \\ 1 & 1 & 2 & 8620 \end{array} |$

خطوة 1.7

The minor for $a_{13}$ is the determinant with row $1$ and column $3$ deleted.

$| \begin{array}{c} 0 & 0 & 1 & 1500 \\ 1 & 0 & 0 & 1000 \\ 0 & 1 & 1 & 2000 \\ 1 & 4 & 2 & 8620 \end{array} |$

خطوة 1.8

Multiply element $a_{13}$ by its cofactor.

$0 | \begin{array}{c} 0 & 0 & 1 & 1500 \\ 1 & 0 & 0 & 1000 \\ 0 & 1 & 1 & 2000 \\ 1 & 4 & 2 & 8620 \end{array} |$

خطوة 1.9

The minor for $a_{14}$ is the determinant with row $1$ and column $4$ deleted.

$| \begin{array}{c} 0 & 0 & 1 & 1500 \\ 1 & 0 & 1 & 1000 \\ 0 & 1 & 0 & 2000 \\ 1 & 4 & 1 & 8620 \end{array} |$

خطوة 1.10

Multiply element $a_{14}$ by its cofactor.

$0 | \begin{array}{c} 0 & 0 & 1 & 1500 \\ 1 & 0 & 1 & 1000 \\ 0 & 1 & 0 & 2000 \\ 1 & 4 & 1 & 8620 \end{array} |$

خطوة 1.11

The minor for $a_{15}$ is the determinant with row $1$ and column $5$ deleted.

$| \begin{array}{c} 0 & 0 & 1 & 1 \\ 1 & 0 & 1 & 0 \\ 0 & 1 & 0 & 1 \\ 1 & 4 & 1 & 2 \end{array} |$

خطوة 1.12

Multiply element $a_{15}$ by its cofactor.

$1500 | \begin{array}{c} 0 & 0 & 1 & 1 \\ 1 & 0 & 1 & 0 \\ 0 & 1 & 0 & 1 \\ 1 & 4 & 1 & 2 \end{array} |$

خطوة 1.13

Add the terms together.

$1 | \begin{array}{c} 0 & 1 & 1 & 1500 \\ 0 & 1 & 0 & 1000 \\ 1 & 0 & 1 & 2000 \\ 4 & 1 & 2 & 8620 \end{array} | - 1 | \begin{array}{c} 0 & 1 & 1 & 1500 \\ 1 & 1 & 0 & 1000 \\ 0 & 0 & 1 & 2000 \\ 1 & 1 & 2 & 8620 \end{array} | + 0 | \begin{array}{c} 0 & 0 & 1 & 1500 \\ 1 & 0 & 0 & 1000 \\ 0 & 1 & 1 & 2000 \\ 1 & 4 & 2 & 8620 \end{array} | + 0 | \begin{array}{c} 0 & 0 & 1 & 1500 \\ 1 & 0 & 1 & 1000 \\ 0 & 1 & 0 & 2000 \\ 1 & 4 & 1 & 8620 \end{array} | + 1500 | \begin{array}{c} 0 & 0 & 1 & 1 \\ 1 & 0 & 1 & 0 \\ 0 & 1 & 0 & 1 \\ 1 & 4 & 1 & 2 \end{array} |$

خطوة 2

اضرب $0$ في $| \begin{array}{c} 0 & 0 & 1 & 1500 \\ 1 & 0 & 0 & 1000 \\ 0 & 1 & 1 & 2000 \\ 1 & 4 & 2 & 8620 \end{array} |$ .

$1 | \begin{array}{c} 0 & 1 & 1 & 1500 \\ 0 & 1 & 0 & 1000 \\ 1 & 0 & 1 & 2000 \\ 4 & 1 & 2 & 8620 \end{array} | - 1 | \begin{array}{c} 0 & 1 & 1 & 1500 \\ 1 & 1 & 0 & 1000 \\ 0 & 0 & 1 & 2000 \\ 1 & 1 & 2 & 8620 \end{array} | + 0 + 0 | \begin{array}{c} 0 & 0 & 1 & 1500 \\ 1 & 0 & 1 & 1000 \\ 0 & 1 & 0 & 2000 \\ 1 & 4 & 1 & 8620 \end{array} | + 1500 | \begin{array}{c} 0 & 0 & 1 & 1 \\ 1 & 0 & 1 & 0 \\ 0 & 1 & 0 & 1 \\ 1 & 4 & 1 & 2 \end{array} |$

خطوة 3

اضرب $0$ في $| \begin{array}{c} 0 & 0 & 1 & 1500 \\ 1 & 0 & 1 & 1000 \\ 0 & 1 & 0 & 2000 \\ 1 & 4 & 1 & 8620 \end{array} |$ .

خطوة 4

احسِب قيمة $| \begin{array}{c} 0 & 1 & 1 & 1500 \\ 0 & 1 & 0 & 1000 \\ 1 & 0 & 1 & 2000 \\ 4 & 1 & 2 & 8620 \end{array} |$ .

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خطوة 4.1

Choose the row or column with the most $0$ elements. If there are no $0$ elements choose any row or column. Multiply every element in column $1$ by its cofactor and add.

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خطوة 4.1.1

Consider the corresponding sign chart.

$| \begin{array}{c} + & - & + & - \\ - & + & - & + \\ + & - & + & - \\ - & + & - & + \end{array} |$

خطوة 4.1.2

The cofactor is the minor with the sign changed if the indices match a $-$ position on the sign chart.

خطوة 4.1.3

The minor for $a_{11}$ is the determinant with row $1$ and column $1$ deleted.

$| \begin{array}{c} 1 & 0 & 1000 \\ 0 & 1 & 2000 \\ 1 & 2 & 8620 \end{array} |$

خطوة 4.1.4

Multiply element $a_{11}$ by its cofactor.

$0 | \begin{array}{c} 1 & 0 & 1000 \\ 0 & 1 & 2000 \\ 1 & 2 & 8620 \end{array} |$

خطوة 4.1.5

The minor for $a_{21}$ is the determinant with row $2$ and column $1$ deleted.

$| \begin{array}{c} 1 & 1 & 1500 \\ 0 & 1 & 2000 \\ 1 & 2 & 8620 \end{array} |$

خطوة 4.1.6

Multiply element $a_{21}$ by its cofactor.

$0 | \begin{array}{c} 1 & 1 & 1500 \\ 0 & 1 & 2000 \\ 1 & 2 & 8620 \end{array} |$

خطوة 4.1.7

The minor for $a_{31}$ is the determinant with row $3$ and column $1$ deleted.

$| \begin{array}{c} 1 & 1 & 1500 \\ 1 & 0 & 1000 \\ 1 & 2 & 8620 \end{array} |$

خطوة 4.1.8

Multiply element $a_{31}$ by its cofactor.

$1 | \begin{array}{c} 1 & 1 & 1500 \\ 1 & 0 & 1000 \\ 1 & 2 & 8620 \end{array} |$

خطوة 4.1.9

The minor for $a_{41}$ is the determinant with row $4$ and column $1$ deleted.

$| \begin{array}{c} 1 & 1 & 1500 \\ 1 & 0 & 1000 \\ 0 & 1 & 2000 \end{array} |$

خطوة 4.1.10

Multiply element $a_{41}$ by its cofactor.

$- 4 | \begin{array}{c} 1 & 1 & 1500 \\ 1 & 0 & 1000 \\ 0 & 1 & 2000 \end{array} |$

خطوة 4.1.11

Add the terms together.

$1 (0 | \begin{array}{c} 1 & 0 & 1000 \\ 0 & 1 & 2000 \\ 1 & 2 & 8620 \end{array} | + 0 | \begin{array}{c} 1 & 1 & 1500 \\ 0 & 1 & 2000 \\ 1 & 2 & 8620 \end{array} | + 1 | \begin{array}{c} 1 & 1 & 1500 \\ 1 & 0 & 1000 \\ 1 & 2 & 8620 \end{array} | - 4 | \begin{array}{c} 1 & 1 & 1500 \\ 1 & 0 & 1000 \\ 0 & 1 & 2000 \end{array} |) - 1 | \begin{array}{c} 0 & 1 & 1 & 1500 \\ 1 & 1 & 0 & 1000 \\ 0 & 0 & 1 & 2000 \\ 1 & 1 & 2 & 8620 \end{array} | + 0 + 0 + 1500 | \begin{array}{c} 0 & 0 & 1 & 1 \\ 1 & 0 & 1 & 0 \\ 0 & 1 & 0 & 1 \\ 1 & 4 & 1 & 2 \end{array} |$

خطوة 4.2

اضرب $0$ في $| \begin{array}{c} 1 & 0 & 1000 \\ 0 & 1 & 2000 \\ 1 & 2 & 8620 \end{array} |$ .

$1 (0 + 0 | \begin{array}{c} 1 & 1 & 1500 \\ 0 & 1 & 2000 \\ 1 & 2 & 8620 \end{array} | + 1 | \begin{array}{c} 1 & 1 & 1500 \\ 1 & 0 & 1000 \\ 1 & 2 & 8620 \end{array} | - 4 | \begin{array}{c} 1 & 1 & 1500 \\ 1 & 0 & 1000 \\ 0 & 1 & 2000 \end{array} |) - 1 | \begin{array}{c} 0 & 1 & 1 & 1500 \\ 1 & 1 & 0 & 1000 \\ 0 & 0 & 1 & 2000 \\ 1 & 1 & 2 & 8620 \end{array} | + 0 + 0 + 1500 | \begin{array}{c} 0 & 0 & 1 & 1 \\ 1 & 0 & 1 & 0 \\ 0 & 1 & 0 & 1 \\ 1 & 4 & 1 & 2 \end{array} |$

خطوة 4.3

اضرب $0$ في $| \begin{array}{c} 1 & 1 & 1500 \\ 0 & 1 & 2000 \\ 1 & 2 & 8620 \end{array} |$ .

$1 (0 + 0 + 1 | \begin{array}{c} 1 & 1 & 1500 \\ 1 & 0 & 1000 \\ 1 & 2 & 8620 \end{array} | - 4 | \begin{array}{c} 1 & 1 & 1500 \\ 1 & 0 & 1000 \\ 0 & 1 & 2000 \end{array} |) - 1 | \begin{array}{c} 0 & 1 & 1 & 1500 \\ 1 & 1 & 0 & 1000 \\ 0 & 0 & 1 & 2000 \\ 1 & 1 & 2 & 8620 \end{array} | + 0 + 0 + 1500 | \begin{array}{c} 0 & 0 & 1 & 1 \\ 1 & 0 & 1 & 0 \\ 0 & 1 & 0 & 1 \\ 1 & 4 & 1 & 2 \end{array} |$

خطوة 4.4

احسِب قيمة $| \begin{array}{c} 1 & 1 & 1500 \\ 1 & 0 & 1000 \\ 1 & 2 & 8620 \end{array} |$ .

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خطوة 4.4.1

Choose the row or column with the most $0$ elements. If there are no $0$ elements choose any row or column. Multiply every element in row $2$ by its cofactor and add.

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خطوة 4.4.1.1

Consider the corresponding sign chart.

$| \begin{array}{c} + & - & + \\ - & + & - \\ + & - & + \end{array} |$

خطوة 4.4.1.2

The cofactor is the minor with the sign changed if the indices match a $-$ position on the sign chart.

خطوة 4.4.1.3

The minor for $a_{21}$ is the determinant with row $2$ and column $1$ deleted.

$| \begin{array}{c} 1 & 1500 \\ 2 & 8620 \end{array} |$

خطوة 4.4.1.4

Multiply element $a_{21}$ by its cofactor.

$- 1 | \begin{array}{c} 1 & 1500 \\ 2 & 8620 \end{array} |$

خطوة 4.4.1.5

The minor for $a_{22}$ is the determinant with row $2$ and column $2$ deleted.

$| \begin{array}{c} 1 & 1500 \\ 1 & 8620 \end{array} |$

خطوة 4.4.1.6

Multiply element $a_{22}$ by its cofactor.

$0 | \begin{array}{c} 1 & 1500 \\ 1 & 8620 \end{array} |$

خطوة 4.4.1.7

The minor for $a_{23}$ is the determinant with row $2$ and column $3$ deleted.

$| \begin{array}{c} 1 & 1 \\ 1 & 2 \end{array} |$

خطوة 4.4.1.8

Multiply element $a_{23}$ by its cofactor.

$- 1000 | \begin{array}{c} 1 & 1 \\ 1 & 2 \end{array} |$

خطوة 4.4.1.9

Add the terms together.

$1 (0 + 0 + 1 (- 1 | \begin{array}{c} 1 & 1500 \\ 2 & 8620 \end{array} | + 0 | \begin{array}{c} 1 & 1500 \\ 1 & 8620 \end{array} | - 1000 | \begin{array}{c} 1 & 1 \\ 1 & 2 \end{array} |) - 4 | \begin{array}{c} 1 & 1 & 1500 \\ 1 & 0 & 1000 \\ 0 & 1 & 2000 \end{array} |) - 1 | \begin{array}{c} 0 & 1 & 1 & 1500 \\ 1 & 1 & 0 & 1000 \\ 0 & 0 & 1 & 2000 \\ 1 & 1 & 2 & 8620 \end{array} | + 0 + 0 + 1500 | \begin{array}{c} 0 & 0 & 1 & 1 \\ 1 & 0 & 1 & 0 \\ 0 & 1 & 0 & 1 \\ 1 & 4 & 1 & 2 \end{array} |$

خطوة 4.4.2

اضرب $0$ في $| \begin{array}{c} 1 & 1500 \\ 1 & 8620 \end{array} |$ .

$1 (0 + 0 + 1 (- 1 | \begin{array}{c} 1 & 1500 \\ 2 & 8620 \end{array} | + 0 - 1000 | \begin{array}{c} 1 & 1 \\ 1 & 2 \end{array} |) - 4 | \begin{array}{c} 1 & 1 & 1500 \\ 1 & 0 & 1000 \\ 0 & 1 & 2000 \end{array} |) - 1 | \begin{array}{c} 0 & 1 & 1 & 1500 \\ 1 & 1 & 0 & 1000 \\ 0 & 0 & 1 & 2000 \\ 1 & 1 & 2 & 8620 \end{array} | + 0 + 0 + 1500 | \begin{array}{c} 0 & 0 & 1 & 1 \\ 1 & 0 & 1 & 0 \\ 0 & 1 & 0 & 1 \\ 1 & 4 & 1 & 2 \end{array} |$

خطوة 4.4.3

احسِب قيمة $| \begin{array}{c} 1 & 1500 \\ 2 & 8620 \end{array} |$ .

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خطوة 4.4.3.1

يمكن إيجاد محدد المصفوفة $2 \times 2$ باستخدام القاعدة $| \begin{array}{c} a & b \\ c & d \end{array} | = a d - c b$ .

$1 (0 + 0 + 1 (- 1 (1 \cdot 8620 - 2 \cdot 1500) + 0 - 1000 | \begin{array}{c} 1 & 1 \\ 1 & 2 \end{array} |) - 4 | \begin{array}{c} 1 & 1 & 1500 \\ 1 & 0 & 1000 \\ 0 & 1 & 2000 \end{array} |) - 1 | \begin{array}{c} 0 & 1 & 1 & 1500 \\ 1 & 1 & 0 & 1000 \\ 0 & 0 & 1 & 2000 \\ 1 & 1 & 2 & 8620 \end{array} | + 0 + 0 + 1500 | \begin{array}{c} 0 & 0 & 1 & 1 \\ 1 & 0 & 1 & 0 \\ 0 & 1 & 0 & 1 \\ 1 & 4 & 1 & 2 \end{array} |$

خطوة 4.4.3.2

بسّط المحدد.

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خطوة 4.4.3.2.1

بسّط كل حد.

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خطوة 4.4.3.2.1.1

اضرب $8620$ في $1$ .

$1 (0 + 0 + 1 (- 1 (8620 - 2 \cdot 1500) + 0 - 1000 | \begin{array}{c} 1 & 1 \\ 1 & 2 \end{array} |) - 4 | \begin{array}{c} 1 & 1 & 1500 \\ 1 & 0 & 1000 \\ 0 & 1 & 2000 \end{array} |) - 1 | \begin{array}{c} 0 & 1 & 1 & 1500 \\ 1 & 1 & 0 & 1000 \\ 0 & 0 & 1 & 2000 \\ 1 & 1 & 2 & 8620 \end{array} | + 0 + 0 + 1500 | \begin{array}{c} 0 & 0 & 1 & 1 \\ 1 & 0 & 1 & 0 \\ 0 & 1 & 0 & 1 \\ 1 & 4 & 1 & 2 \end{array} |$

خطوة 4.4.3.2.1.2

اضرب $- 2$ في $1500$ .

$1 (0 + 0 + 1 (- 1 (8620 - 3000) + 0 - 1000 | \begin{array}{c} 1 & 1 \\ 1 & 2 \end{array} |) - 4 | \begin{array}{c} 1 & 1 & 1500 \\ 1 & 0 & 1000 \\ 0 & 1 & 2000 \end{array} |) - 1 | \begin{array}{c} 0 & 1 & 1 & 1500 \\ 1 & 1 & 0 & 1000 \\ 0 & 0 & 1 & 2000 \\ 1 & 1 & 2 & 8620 \end{array} | + 0 + 0 + 1500 | \begin{array}{c} 0 & 0 & 1 & 1 \\ 1 & 0 & 1 & 0 \\ 0 & 1 & 0 & 1 \\ 1 & 4 & 1 & 2 \end{array} |$

خطوة 4.4.3.2.2

اطرح $3000$ من $8620$ .

$1 (0 + 0 + 1 (- 1 \cdot 5620 + 0 - 1000 | \begin{array}{c} 1 & 1 \\ 1 & 2 \end{array} |) - 4 | \begin{array}{c} 1 & 1 & 1500 \\ 1 & 0 & 1000 \\ 0 & 1 & 2000 \end{array} |) - 1 | \begin{array}{c} 0 & 1 & 1 & 1500 \\ 1 & 1 & 0 & 1000 \\ 0 & 0 & 1 & 2000 \\ 1 & 1 & 2 & 8620 \end{array} | + 0 + 0 + 1500 | \begin{array}{c} 0 & 0 & 1 & 1 \\ 1 & 0 & 1 & 0 \\ 0 & 1 & 0 & 1 \\ 1 & 4 & 1 & 2 \end{array} |$

خطوة 4.4.4

احسِب قيمة $| \begin{array}{c} 1 & 1 \\ 1 & 2 \end{array} |$ .

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خطوة 4.4.4.1

يمكن إيجاد محدد المصفوفة $2 \times 2$ باستخدام القاعدة $| \begin{array}{c} a & b \\ c & d \end{array} | = a d - c b$ .

$1 (0 + 0 + 1 (- 1 \cdot 5620 + 0 - 1000 (1 \cdot 2 - 1 \cdot 1)) - 4 | \begin{array}{c} 1 & 1 & 1500 \\ 1 & 0 & 1000 \\ 0 & 1 & 2000 \end{array} |) - 1 | \begin{array}{c} 0 & 1 & 1 & 1500 \\ 1 & 1 & 0 & 1000 \\ 0 & 0 & 1 & 2000 \\ 1 & 1 & 2 & 8620 \end{array} | + 0 + 0 + 1500 | \begin{array}{c} 0 & 0 & 1 & 1 \\ 1 & 0 & 1 & 0 \\ 0 & 1 & 0 & 1 \\ 1 & 4 & 1 & 2 \end{array} |$

خطوة 4.4.4.2

بسّط المحدد.

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خطوة 4.4.4.2.1

بسّط كل حد.

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خطوة 4.4.4.2.1.1

اضرب $2$ في $1$ .

$1 (0 + 0 + 1 (- 1 \cdot 5620 + 0 - 1000 (2 - 1 \cdot 1)) - 4 | \begin{array}{c} 1 & 1 & 1500 \\ 1 & 0 & 1000 \\ 0 & 1 & 2000 \end{array} |) - 1 | \begin{array}{c} 0 & 1 & 1 & 1500 \\ 1 & 1 & 0 & 1000 \\ 0 & 0 & 1 & 2000 \\ 1 & 1 & 2 & 8620 \end{array} | + 0 + 0 + 1500 | \begin{array}{c} 0 & 0 & 1 & 1 \\ 1 & 0 & 1 & 0 \\ 0 & 1 & 0 & 1 \\ 1 & 4 & 1 & 2 \end{array} |$

خطوة 4.4.4.2.1.2

اضرب $- 1$ في $1$ .

$1 (0 + 0 + 1 (- 1 \cdot 5620 + 0 - 1000 (2 - 1)) - 4 | \begin{array}{c} 1 & 1 & 1500 \\ 1 & 0 & 1000 \\ 0 & 1 & 2000 \end{array} |) - 1 | \begin{array}{c} 0 & 1 & 1 & 1500 \\ 1 & 1 & 0 & 1000 \\ 0 & 0 & 1 & 2000 \\ 1 & 1 & 2 & 8620 \end{array} | + 0 + 0 + 1500 | \begin{array}{c} 0 & 0 & 1 & 1 \\ 1 & 0 & 1 & 0 \\ 0 & 1 & 0 & 1 \\ 1 & 4 & 1 & 2 \end{array} |$

خطوة 4.4.4.2.2

اطرح $1$ من $2$ .

$1 (0 + 0 + 1 (- 1 \cdot 5620 + 0 - 1000 \cdot 1) - 4 | \begin{array}{c} 1 & 1 & 1500 \\ 1 & 0 & 1000 \\ 0 & 1 & 2000 \end{array} |) - 1 | \begin{array}{c} 0 & 1 & 1 & 1500 \\ 1 & 1 & 0 & 1000 \\ 0 & 0 & 1 & 2000 \\ 1 & 1 & 2 & 8620 \end{array} | + 0 + 0 + 1500 | \begin{array}{c} 0 & 0 & 1 & 1 \\ 1 & 0 & 1 & 0 \\ 0 & 1 & 0 & 1 \\ 1 & 4 & 1 & 2 \end{array} |$

خطوة 4.4.5

بسّط المحدد.

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خطوة 4.4.5.1

بسّط كل حد.

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خطوة 4.4.5.1.1

اضرب $- 1$ في $5620$ .

$1 (0 + 0 + 1 (- 5620 + 0 - 1000 \cdot 1) - 4 | \begin{array}{c} 1 & 1 & 1500 \\ 1 & 0 & 1000 \\ 0 & 1 & 2000 \end{array} |) - 1 | \begin{array}{c} 0 & 1 & 1 & 1500 \\ 1 & 1 & 0 & 1000 \\ 0 & 0 & 1 & 2000 \\ 1 & 1 & 2 & 8620 \end{array} | + 0 + 0 + 1500 | \begin{array}{c} 0 & 0 & 1 & 1 \\ 1 & 0 & 1 & 0 \\ 0 & 1 & 0 & 1 \\ 1 & 4 & 1 & 2 \end{array} |$

خطوة 4.4.5.1.2

اضرب $- 1000$ في $1$ .

$1 (0 + 0 + 1 (- 5620 + 0 - 1000) - 4 | \begin{array}{c} 1 & 1 & 1500 \\ 1 & 0 & 1000 \\ 0 & 1 & 2000 \end{array} |) - 1 | \begin{array}{c} 0 & 1 & 1 & 1500 \\ 1 & 1 & 0 & 1000 \\ 0 & 0 & 1 & 2000 \\ 1 & 1 & 2 & 8620 \end{array} | + 0 + 0 + 1500 | \begin{array}{c} 0 & 0 & 1 & 1 \\ 1 & 0 & 1 & 0 \\ 0 & 1 & 0 & 1 \\ 1 & 4 & 1 & 2 \end{array} |$

خطوة 4.4.5.2

أضف $- 5620$ و $0$ .

$1 (0 + 0 + 1 (- 5620 - 1000) - 4 | \begin{array}{c} 1 & 1 & 1500 \\ 1 & 0 & 1000 \\ 0 & 1 & 2000 \end{array} |) - 1 | \begin{array}{c} 0 & 1 & 1 & 1500 \\ 1 & 1 & 0 & 1000 \\ 0 & 0 & 1 & 2000 \\ 1 & 1 & 2 & 8620 \end{array} | + 0 + 0 + 1500 | \begin{array}{c} 0 & 0 & 1 & 1 \\ 1 & 0 & 1 & 0 \\ 0 & 1 & 0 & 1 \\ 1 & 4 & 1 & 2 \end{array} |$

خطوة 4.4.5.3

اطرح $1000$ من $- 5620$ .

$1 (0 + 0 + 1 \cdot - 6620 - 4 | \begin{array}{c} 1 & 1 & 1500 \\ 1 & 0 & 1000 \\ 0 & 1 & 2000 \end{array} |) - 1 | \begin{array}{c} 0 & 1 & 1 & 1500 \\ 1 & 1 & 0 & 1000 \\ 0 & 0 & 1 & 2000 \\ 1 & 1 & 2 & 8620 \end{array} | + 0 + 0 + 1500 | \begin{array}{c} 0 & 0 & 1 & 1 \\ 1 & 0 & 1 & 0 \\ 0 & 1 & 0 & 1 \\ 1 & 4 & 1 & 2 \end{array} |$

خطوة 4.5

احسِب قيمة $| \begin{array}{c} 1 & 1 & 1500 \\ 1 & 0 & 1000 \\ 0 & 1 & 2000 \end{array} |$ .

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خطوة 4.5.1

Choose the row or column with the most $0$ elements. If there are no $0$ elements choose any row or column. Multiply every element in column $1$ by its cofactor and add.

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خطوة 4.5.1.1

Consider the corresponding sign chart.

$| \begin{array}{c} + & - & + \\ - & + & - \\ + & - & + \end{array} |$

خطوة 4.5.1.2

The cofactor is the minor with the sign changed if the indices match a $-$ position on the sign chart.

خطوة 4.5.1.3

The minor for $a_{11}$ is the determinant with row $1$ and column $1$ deleted.

$| \begin{array}{c} 0 & 1000 \\ 1 & 2000 \end{array} |$

خطوة 4.5.1.4

Multiply element $a_{11}$ by its cofactor.

$1 | \begin{array}{c} 0 & 1000 \\ 1 & 2000 \end{array} |$

خطوة 4.5.1.5

The minor for $a_{21}$ is the determinant with row $2$ and column $1$ deleted.

$| \begin{array}{c} 1 & 1500 \\ 1 & 2000 \end{array} |$

خطوة 4.5.1.6

Multiply element $a_{21}$ by its cofactor.

$- 1 | \begin{array}{c} 1 & 1500 \\ 1 & 2000 \end{array} |$

خطوة 4.5.1.7

The minor for $a_{31}$ is the determinant with row $3$ and column $1$ deleted.

$| \begin{array}{c} 1 & 1500 \\ 0 & 1000 \end{array} |$

خطوة 4.5.1.8

Multiply element $a_{31}$ by its cofactor.

$0 | \begin{array}{c} 1 & 1500 \\ 0 & 1000 \end{array} |$

خطوة 4.5.1.9

Add the terms together.

$1 (0 + 0 + 1 \cdot - 6620 - 4 (1 | \begin{array}{c} 0 & 1000 \\ 1 & 2000 \end{array} | - 1 | \begin{array}{c} 1 & 1500 \\ 1 & 2000 \end{array} | + 0 | \begin{array}{c} 1 & 1500 \\ 0 & 1000 \end{array} |)) - 1 | \begin{array}{c} 0 & 1 & 1 & 1500 \\ 1 & 1 & 0 & 1000 \\ 0 & 0 & 1 & 2000 \\ 1 & 1 & 2 & 8620 \end{array} | + 0 + 0 + 1500 | \begin{array}{c} 0 & 0 & 1 & 1 \\ 1 & 0 & 1 & 0 \\ 0 & 1 & 0 & 1 \\ 1 & 4 & 1 & 2 \end{array} |$

خطوة 4.5.2

اضرب $0$ في $| \begin{array}{c} 1 & 1500 \\ 0 & 1000 \end{array} |$ .

$1 (0 + 0 + 1 \cdot - 6620 - 4 (1 | \begin{array}{c} 0 & 1000 \\ 1 & 2000 \end{array} | - 1 | \begin{array}{c} 1 & 1500 \\ 1 & 2000 \end{array} | + 0)) - 1 | \begin{array}{c} 0 & 1 & 1 & 1500 \\ 1 & 1 & 0 & 1000 \\ 0 & 0 & 1 & 2000 \\ 1 & 1 & 2 & 8620 \end{array} | + 0 + 0 + 1500 | \begin{array}{c} 0 & 0 & 1 & 1 \\ 1 & 0 & 1 & 0 \\ 0 & 1 & 0 & 1 \\ 1 & 4 & 1 & 2 \end{array} |$

خطوة 4.5.3

احسِب قيمة $| \begin{array}{c} 0 & 1000 \\ 1 & 2000 \end{array} |$ .

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خطوة 4.5.3.1

يمكن إيجاد محدد المصفوفة $2 \times 2$ باستخدام القاعدة $| \begin{array}{c} a & b \\ c & d \end{array} | = a d - c b$ .

$1 (0 + 0 + 1 \cdot - 6620 - 4 (1 (0 \cdot 2000 - 1 \cdot 1000) - 1 | \begin{array}{c} 1 & 1500 \\ 1 & 2000 \end{array} | + 0)) - 1 | \begin{array}{c} 0 & 1 & 1 & 1500 \\ 1 & 1 & 0 & 1000 \\ 0 & 0 & 1 & 2000 \\ 1 & 1 & 2 & 8620 \end{array} | + 0 + 0 + 1500 | \begin{array}{c} 0 & 0 & 1 & 1 \\ 1 & 0 & 1 & 0 \\ 0 & 1 & 0 & 1 \\ 1 & 4 & 1 & 2 \end{array} |$

خطوة 4.5.3.2

بسّط المحدد.

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خطوة 4.5.3.2.1

بسّط كل حد.

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خطوة 4.5.3.2.1.1

اضرب $0$ في $2000$ .

$1 (0 + 0 + 1 \cdot - 6620 - 4 (1 (0 - 1 \cdot 1000) - 1 | \begin{array}{c} 1 & 1500 \\ 1 & 2000 \end{array} | + 0)) - 1 | \begin{array}{c} 0 & 1 & 1 & 1500 \\ 1 & 1 & 0 & 1000 \\ 0 & 0 & 1 & 2000 \\ 1 & 1 & 2 & 8620 \end{array} | + 0 + 0 + 1500 | \begin{array}{c} 0 & 0 & 1 & 1 \\ 1 & 0 & 1 & 0 \\ 0 & 1 & 0 & 1 \\ 1 & 4 & 1 & 2 \end{array} |$

خطوة 4.5.3.2.1.2

اضرب $- 1$ في $1000$ .

$1 (0 + 0 + 1 \cdot - 6620 - 4 (1 (0 - 1000) - 1 | \begin{array}{c} 1 & 1500 \\ 1 & 2000 \end{array} | + 0)) - 1 | \begin{array}{c} 0 & 1 & 1 & 1500 \\ 1 & 1 & 0 & 1000 \\ 0 & 0 & 1 & 2000 \\ 1 & 1 & 2 & 8620 \end{array} | + 0 + 0 + 1500 | \begin{array}{c} 0 & 0 & 1 & 1 \\ 1 & 0 & 1 & 0 \\ 0 & 1 & 0 & 1 \\ 1 & 4 & 1 & 2 \end{array} |$

خطوة 4.5.3.2.2

اطرح $1000$ من $0$ .

$1 (0 + 0 + 1 \cdot - 6620 - 4 (1 \cdot - 1000 - 1 | \begin{array}{c} 1 & 1500 \\ 1 & 2000 \end{array} | + 0)) - 1 | \begin{array}{c} 0 & 1 & 1 & 1500 \\ 1 & 1 & 0 & 1000 \\ 0 & 0 & 1 & 2000 \\ 1 & 1 & 2 & 8620 \end{array} | + 0 + 0 + 1500 | \begin{array}{c} 0 & 0 & 1 & 1 \\ 1 & 0 & 1 & 0 \\ 0 & 1 & 0 & 1 \\ 1 & 4 & 1 & 2 \end{array} |$

خطوة 4.5.4

احسِب قيمة $| \begin{array}{c} 1 & 1500 \\ 1 & 2000 \end{array} |$ .

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خطوة 4.5.4.1

يمكن إيجاد محدد المصفوفة $2 \times 2$ باستخدام القاعدة $| \begin{array}{c} a & b \\ c & d \end{array} | = a d - c b$ .

$1 (0 + 0 + 1 \cdot - 6620 - 4 (1 \cdot - 1000 - 1 (1 \cdot 2000 - 1 \cdot 1500) + 0)) - 1 | \begin{array}{c} 0 & 1 & 1 & 1500 \\ 1 & 1 & 0 & 1000 \\ 0 & 0 & 1 & 2000 \\ 1 & 1 & 2 & 8620 \end{array} | + 0 + 0 + 1500 | \begin{array}{c} 0 & 0 & 1 & 1 \\ 1 & 0 & 1 & 0 \\ 0 & 1 & 0 & 1 \\ 1 & 4 & 1 & 2 \end{array} |$

خطوة 4.5.4.2

بسّط المحدد.

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خطوة 4.5.4.2.1

بسّط كل حد.

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خطوة 4.5.4.2.1.1

اضرب $2000$ في $1$ .

$1 (0 + 0 + 1 \cdot - 6620 - 4 (1 \cdot - 1000 - 1 (2000 - 1 \cdot 1500) + 0)) - 1 | \begin{array}{c} 0 & 1 & 1 & 1500 \\ 1 & 1 & 0 & 1000 \\ 0 & 0 & 1 & 2000 \\ 1 & 1 & 2 & 8620 \end{array} | + 0 + 0 + 1500 | \begin{array}{c} 0 & 0 & 1 & 1 \\ 1 & 0 & 1 & 0 \\ 0 & 1 & 0 & 1 \\ 1 & 4 & 1 & 2 \end{array} |$

خطوة 4.5.4.2.1.2

اضرب $- 1$ في $1500$ .

$1 (0 + 0 + 1 \cdot - 6620 - 4 (1 \cdot - 1000 - 1 (2000 - 1500) + 0)) - 1 | \begin{array}{c} 0 & 1 & 1 & 1500 \\ 1 & 1 & 0 & 1000 \\ 0 & 0 & 1 & 2000 \\ 1 & 1 & 2 & 8620 \end{array} | + 0 + 0 + 1500 | \begin{array}{c} 0 & 0 & 1 & 1 \\ 1 & 0 & 1 & 0 \\ 0 & 1 & 0 & 1 \\ 1 & 4 & 1 & 2 \end{array} |$

خطوة 4.5.4.2.2

اطرح $1500$ من $2000$ .

$1 (0 + 0 + 1 \cdot - 6620 - 4 (1 \cdot - 1000 - 1 \cdot 500 + 0)) - 1 | \begin{array}{c} 0 & 1 & 1 & 1500 \\ 1 & 1 & 0 & 1000 \\ 0 & 0 & 1 & 2000 \\ 1 & 1 & 2 & 8620 \end{array} | + 0 + 0 + 1500 | \begin{array}{c} 0 & 0 & 1 & 1 \\ 1 & 0 & 1 & 0 \\ 0 & 1 & 0 & 1 \\ 1 & 4 & 1 & 2 \end{array} |$

خطوة 4.5.5

بسّط المحدد.

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خطوة 4.5.5.1

بسّط كل حد.

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خطوة 4.5.5.1.1

اضرب $- 1000$ في $1$ .

$1 (0 + 0 + 1 \cdot - 6620 - 4 (- 1000 - 1 \cdot 500 + 0)) - 1 | \begin{array}{c} 0 & 1 & 1 & 1500 \\ 1 & 1 & 0 & 1000 \\ 0 & 0 & 1 & 2000 \\ 1 & 1 & 2 & 8620 \end{array} | + 0 + 0 + 1500 | \begin{array}{c} 0 & 0 & 1 & 1 \\ 1 & 0 & 1 & 0 \\ 0 & 1 & 0 & 1 \\ 1 & 4 & 1 & 2 \end{array} |$

خطوة 4.5.5.1.2

اضرب $- 1$ في $500$ .

$1 (0 + 0 + 1 \cdot - 6620 - 4 (- 1000 - 500 + 0)) - 1 | \begin{array}{c} 0 & 1 & 1 & 1500 \\ 1 & 1 & 0 & 1000 \\ 0 & 0 & 1 & 2000 \\ 1 & 1 & 2 & 8620 \end{array} | + 0 + 0 + 1500 | \begin{array}{c} 0 & 0 & 1 & 1 \\ 1 & 0 & 1 & 0 \\ 0 & 1 & 0 & 1 \\ 1 & 4 & 1 & 2 \end{array} |$

خطوة 4.5.5.2

اطرح $500$ من $- 1000$ .

$1 (0 + 0 + 1 \cdot - 6620 - 4 (- 1500 + 0)) - 1 | \begin{array}{c} 0 & 1 & 1 & 1500 \\ 1 & 1 & 0 & 1000 \\ 0 & 0 & 1 & 2000 \\ 1 & 1 & 2 & 8620 \end{array} | + 0 + 0 + 1500 | \begin{array}{c} 0 & 0 & 1 & 1 \\ 1 & 0 & 1 & 0 \\ 0 & 1 & 0 & 1 \\ 1 & 4 & 1 & 2 \end{array} |$

خطوة 4.5.5.3

أضف $- 1500$ و $0$ .

$1 (0 + 0 + 1 \cdot - 6620 - 4 \cdot - 1500) - 1 | \begin{array}{c} 0 & 1 & 1 & 1500 \\ 1 & 1 & 0 & 1000 \\ 0 & 0 & 1 & 2000 \\ 1 & 1 & 2 & 8620 \end{array} | + 0 + 0 + 1500 | \begin{array}{c} 0 & 0 & 1 & 1 \\ 1 & 0 & 1 & 0 \\ 0 & 1 & 0 & 1 \\ 1 & 4 & 1 & 2 \end{array} |$

خطوة 4.6

بسّط المحدد.

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خطوة 4.6.1

بسّط كل حد.

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خطوة 4.6.1.1

اضرب $- 6620$ في $1$ .

$1 (0 + 0 - 6620 - 4 \cdot - 1500) - 1 | \begin{array}{c} 0 & 1 & 1 & 1500 \\ 1 & 1 & 0 & 1000 \\ 0 & 0 & 1 & 2000 \\ 1 & 1 & 2 & 8620 \end{array} | + 0 + 0 + 1500 | \begin{array}{c} 0 & 0 & 1 & 1 \\ 1 & 0 & 1 & 0 \\ 0 & 1 & 0 & 1 \\ 1 & 4 & 1 & 2 \end{array} |$

خطوة 4.6.1.2

اضرب $- 4$ في $- 1500$ .

$1 (0 + 0 - 6620 + 6000) - 1 | \begin{array}{c} 0 & 1 & 1 & 1500 \\ 1 & 1 & 0 & 1000 \\ 0 & 0 & 1 & 2000 \\ 1 & 1 & 2 & 8620 \end{array} | + 0 + 0 + 1500 | \begin{array}{c} 0 & 0 & 1 & 1 \\ 1 & 0 & 1 & 0 \\ 0 & 1 & 0 & 1 \\ 1 & 4 & 1 & 2 \end{array} |$

خطوة 4.6.2

أضف $0$ و $0$ .

$1 (0 - 6620 + 6000) - 1 | \begin{array}{c} 0 & 1 & 1 & 1500 \\ 1 & 1 & 0 & 1000 \\ 0 & 0 & 1 & 2000 \\ 1 & 1 & 2 & 8620 \end{array} | + 0 + 0 + 1500 | \begin{array}{c} 0 & 0 & 1 & 1 \\ 1 & 0 & 1 & 0 \\ 0 & 1 & 0 & 1 \\ 1 & 4 & 1 & 2 \end{array} |$

خطوة 4.6.3

اطرح $6620$ من $0$ .

$1 (- 6620 + 6000) - 1 | \begin{array}{c} 0 & 1 & 1 & 1500 \\ 1 & 1 & 0 & 1000 \\ 0 & 0 & 1 & 2000 \\ 1 & 1 & 2 & 8620 \end{array} | + 0 + 0 + 1500 | \begin{array}{c} 0 & 0 & 1 & 1 \\ 1 & 0 & 1 & 0 \\ 0 & 1 & 0 & 1 \\ 1 & 4 & 1 & 2 \end{array} |$

خطوة 4.6.4

أضف $- 6620$ و $6000$ .

$1 \cdot - 620 - 1 | \begin{array}{c} 0 & 1 & 1 & 1500 \\ 1 & 1 & 0 & 1000 \\ 0 & 0 & 1 & 2000 \\ 1 & 1 & 2 & 8620 \end{array} | + 0 + 0 + 1500 | \begin{array}{c} 0 & 0 & 1 & 1 \\ 1 & 0 & 1 & 0 \\ 0 & 1 & 0 & 1 \\ 1 & 4 & 1 & 2 \end{array} |$

خطوة 5

احسِب قيمة $| \begin{array}{c} 0 & 1 & 1 & 1500 \\ 1 & 1 & 0 & 1000 \\ 0 & 0 & 1 & 2000 \\ 1 & 1 & 2 & 8620 \end{array} |$ .

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خطوة 5.1

Choose the row or column with the most $0$ elements. If there are no $0$ elements choose any row or column. Multiply every element in column $1$ by its cofactor and add.

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خطوة 5.1.1

Consider the corresponding sign chart.

$| \begin{array}{c} + & - & + & - \\ - & + & - & + \\ + & - & + & - \\ - & + & - & + \end{array} |$

خطوة 5.1.2

The cofactor is the minor with the sign changed if the indices match a $-$ position on the sign chart.

خطوة 5.1.3

The minor for $a_{11}$ is the determinant with row $1$ and column $1$ deleted.

$| \begin{array}{c} 1 & 0 & 1000 \\ 0 & 1 & 2000 \\ 1 & 2 & 8620 \end{array} |$

خطوة 5.1.4

Multiply element $a_{11}$ by its cofactor.

$0 | \begin{array}{c} 1 & 0 & 1000 \\ 0 & 1 & 2000 \\ 1 & 2 & 8620 \end{array} |$

خطوة 5.1.5

The minor for $a_{21}$ is the determinant with row $2$ and column $1$ deleted.

$| \begin{array}{c} 1 & 1 & 1500 \\ 0 & 1 & 2000 \\ 1 & 2 & 8620 \end{array} |$

خطوة 5.1.6

Multiply element $a_{21}$ by its cofactor.

$- 1 | \begin{array}{c} 1 & 1 & 1500 \\ 0 & 1 & 2000 \\ 1 & 2 & 8620 \end{array} |$

خطوة 5.1.7

The minor for $a_{31}$ is the determinant with row $3$ and column $1$ deleted.

$| \begin{array}{c} 1 & 1 & 1500 \\ 1 & 0 & 1000 \\ 1 & 2 & 8620 \end{array} |$

خطوة 5.1.8

Multiply element $a_{31}$ by its cofactor.

$0 | \begin{array}{c} 1 & 1 & 1500 \\ 1 & 0 & 1000 \\ 1 & 2 & 8620 \end{array} |$

خطوة 5.1.9

The minor for $a_{41}$ is the determinant with row $4$ and column $1$ deleted.

$| \begin{array}{c} 1 & 1 & 1500 \\ 1 & 0 & 1000 \\ 0 & 1 & 2000 \end{array} |$

خطوة 5.1.10

Multiply element $a_{41}$ by its cofactor.

$- 1 | \begin{array}{c} 1 & 1 & 1500 \\ 1 & 0 & 1000 \\ 0 & 1 & 2000 \end{array} |$

خطوة 5.1.11

Add the terms together.

$1 \cdot - 620 - 1 (0 | \begin{array}{c} 1 & 0 & 1000 \\ 0 & 1 & 2000 \\ 1 & 2 & 8620 \end{array} | - 1 | \begin{array}{c} 1 & 1 & 1500 \\ 0 & 1 & 2000 \\ 1 & 2 & 8620 \end{array} | + 0 | \begin{array}{c} 1 & 1 & 1500 \\ 1 & 0 & 1000 \\ 1 & 2 & 8620 \end{array} | - 1 | \begin{array}{c} 1 & 1 & 1500 \\ 1 & 0 & 1000 \\ 0 & 1 & 2000 \end{array} |) + 0 + 0 + 1500 | \begin{array}{c} 0 & 0 & 1 & 1 \\ 1 & 0 & 1 & 0 \\ 0 & 1 & 0 & 1 \\ 1 & 4 & 1 & 2 \end{array} |$

خطوة 5.2

اضرب $0$ في $| \begin{array}{c} 1 & 0 & 1000 \\ 0 & 1 & 2000 \\ 1 & 2 & 8620 \end{array} |$ .

$1 \cdot - 620 - 1 (0 - 1 | \begin{array}{c} 1 & 1 & 1500 \\ 0 & 1 & 2000 \\ 1 & 2 & 8620 \end{array} | + 0 | \begin{array}{c} 1 & 1 & 1500 \\ 1 & 0 & 1000 \\ 1 & 2 & 8620 \end{array} | - 1 | \begin{array}{c} 1 & 1 & 1500 \\ 1 & 0 & 1000 \\ 0 & 1 & 2000 \end{array} |) + 0 + 0 + 1500 | \begin{array}{c} 0 & 0 & 1 & 1 \\ 1 & 0 & 1 & 0 \\ 0 & 1 & 0 & 1 \\ 1 & 4 & 1 & 2 \end{array} |$

خطوة 5.3

اضرب $0$ في $| \begin{array}{c} 1 & 1 & 1500 \\ 1 & 0 & 1000 \\ 1 & 2 & 8620 \end{array} |$ .

$1 \cdot - 620 - 1 (0 - 1 | \begin{array}{c} 1 & 1 & 1500 \\ 0 & 1 & 2000 \\ 1 & 2 & 8620 \end{array} | + 0 - 1 | \begin{array}{c} 1 & 1 & 1500 \\ 1 & 0 & 1000 \\ 0 & 1 & 2000 \end{array} |) + 0 + 0 + 1500 | \begin{array}{c} 0 & 0 & 1 & 1 \\ 1 & 0 & 1 & 0 \\ 0 & 1 & 0 & 1 \\ 1 & 4 & 1 & 2 \end{array} |$

خطوة 5.4

احسِب قيمة $| \begin{array}{c} 1 & 1 & 1500 \\ 0 & 1 & 2000 \\ 1 & 2 & 8620 \end{array} |$ .

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خطوة 5.4.1

Choose the row or column with the most $0$ elements. If there are no $0$ elements choose any row or column. Multiply every element in column $1$ by its cofactor and add.

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خطوة 5.4.1.1

Consider the corresponding sign chart.

$| \begin{array}{c} + & - & + \\ - & + & - \\ + & - & + \end{array} |$

خطوة 5.4.1.2

The cofactor is the minor with the sign changed if the indices match a $-$ position on the sign chart.

خطوة 5.4.1.3

The minor for $a_{11}$ is the determinant with row $1$ and column $1$ deleted.

$| \begin{array}{c} 1 & 2000 \\ 2 & 8620 \end{array} |$

خطوة 5.4.1.4

Multiply element $a_{11}$ by its cofactor.

$1 | \begin{array}{c} 1 & 2000 \\ 2 & 8620 \end{array} |$

خطوة 5.4.1.5

The minor for $a_{21}$ is the determinant with row $2$ and column $1$ deleted.

$| \begin{array}{c} 1 & 1500 \\ 2 & 8620 \end{array} |$

خطوة 5.4.1.6

Multiply element $a_{21}$ by its cofactor.

$0 | \begin{array}{c} 1 & 1500 \\ 2 & 8620 \end{array} |$

خطوة 5.4.1.7

The minor for $a_{31}$ is the determinant with row $3$ and column $1$ deleted.

$| \begin{array}{c} 1 & 1500 \\ 1 & 2000 \end{array} |$

خطوة 5.4.1.8

Multiply element $a_{31}$ by its cofactor.

$1 | \begin{array}{c} 1 & 1500 \\ 1 & 2000 \end{array} |$

خطوة 5.4.1.9

Add the terms together.

$1 \cdot - 620 - 1 (0 - 1 (1 | \begin{array}{c} 1 & 2000 \\ 2 & 8620 \end{array} | + 0 | \begin{array}{c} 1 & 1500 \\ 2 & 8620 \end{array} | + 1 | \begin{array}{c} 1 & 1500 \\ 1 & 2000 \end{array} |) + 0 - 1 | \begin{array}{c} 1 & 1 & 1500 \\ 1 & 0 & 1000 \\ 0 & 1 & 2000 \end{array} |) + 0 + 0 + 1500 | \begin{array}{c} 0 & 0 & 1 & 1 \\ 1 & 0 & 1 & 0 \\ 0 & 1 & 0 & 1 \\ 1 & 4 & 1 & 2 \end{array} |$

خطوة 5.4.2

اضرب $0$ في $| \begin{array}{c} 1 & 1500 \\ 2 & 8620 \end{array} |$ .

$1 \cdot - 620 - 1 (0 - 1 (1 | \begin{array}{c} 1 & 2000 \\ 2 & 8620 \end{array} | + 0 + 1 | \begin{array}{c} 1 & 1500 \\ 1 & 2000 \end{array} |) + 0 - 1 | \begin{array}{c} 1 & 1 & 1500 \\ 1 & 0 & 1000 \\ 0 & 1 & 2000 \end{array} |) + 0 + 0 + 1500 | \begin{array}{c} 0 & 0 & 1 & 1 \\ 1 & 0 & 1 & 0 \\ 0 & 1 & 0 & 1 \\ 1 & 4 & 1 & 2 \end{array} |$

خطوة 5.4.3

احسِب قيمة $| \begin{array}{c} 1 & 2000 \\ 2 & 8620 \end{array} |$ .

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خطوة 5.4.3.1

يمكن إيجاد محدد المصفوفة $2 \times 2$ باستخدام القاعدة $| \begin{array}{c} a & b \\ c & d \end{array} | = a d - c b$ .

$1 \cdot - 620 - 1 (0 - 1 (1 (1 \cdot 8620 - 2 \cdot 2000) + 0 + 1 | \begin{array}{c} 1 & 1500 \\ 1 & 2000 \end{array} |) + 0 - 1 | \begin{array}{c} 1 & 1 & 1500 \\ 1 & 0 & 1000 \\ 0 & 1 & 2000 \end{array} |) + 0 + 0 + 1500 | \begin{array}{c} 0 & 0 & 1 & 1 \\ 1 & 0 & 1 & 0 \\ 0 & 1 & 0 & 1 \\ 1 & 4 & 1 & 2 \end{array} |$

خطوة 5.4.3.2

بسّط المحدد.

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خطوة 5.4.3.2.1

بسّط كل حد.

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خطوة 5.4.3.2.1.1

اضرب $8620$ في $1$ .

$1 \cdot - 620 - 1 (0 - 1 (1 (8620 - 2 \cdot 2000) + 0 + 1 | \begin{array}{c} 1 & 1500 \\ 1 & 2000 \end{array} |) + 0 - 1 | \begin{array}{c} 1 & 1 & 1500 \\ 1 & 0 & 1000 \\ 0 & 1 & 2000 \end{array} |) + 0 + 0 + 1500 | \begin{array}{c} 0 & 0 & 1 & 1 \\ 1 & 0 & 1 & 0 \\ 0 & 1 & 0 & 1 \\ 1 & 4 & 1 & 2 \end{array} |$

خطوة 5.4.3.2.1.2

اضرب $- 2$ في $2000$ .

$1 \cdot - 620 - 1 (0 - 1 (1 (8620 - 4000) + 0 + 1 | \begin{array}{c} 1 & 1500 \\ 1 & 2000 \end{array} |) + 0 - 1 | \begin{array}{c} 1 & 1 & 1500 \\ 1 & 0 & 1000 \\ 0 & 1 & 2000 \end{array} |) + 0 + 0 + 1500 | \begin{array}{c} 0 & 0 & 1 & 1 \\ 1 & 0 & 1 & 0 \\ 0 & 1 & 0 & 1 \\ 1 & 4 & 1 & 2 \end{array} |$

خطوة 5.4.3.2.2

اطرح $4000$ من $8620$ .

$1 \cdot - 620 - 1 (0 - 1 (1 \cdot 4620 + 0 + 1 | \begin{array}{c} 1 & 1500 \\ 1 & 2000 \end{array} |) + 0 - 1 | \begin{array}{c} 1 & 1 & 1500 \\ 1 & 0 & 1000 \\ 0 & 1 & 2000 \end{array} |) + 0 + 0 + 1500 | \begin{array}{c} 0 & 0 & 1 & 1 \\ 1 & 0 & 1 & 0 \\ 0 & 1 & 0 & 1 \\ 1 & 4 & 1 & 2 \end{array} |$

خطوة 5.4.4

احسِب قيمة $| \begin{array}{c} 1 & 1500 \\ 1 & 2000 \end{array} |$ .

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خطوة 5.4.4.1

يمكن إيجاد محدد المصفوفة $2 \times 2$ باستخدام القاعدة $| \begin{array}{c} a & b \\ c & d \end{array} | = a d - c b$ .

$1 \cdot - 620 - 1 (0 - 1 (1 \cdot 4620 + 0 + 1 (1 \cdot 2000 - 1 \cdot 1500)) + 0 - 1 | \begin{array}{c} 1 & 1 & 1500 \\ 1 & 0 & 1000 \\ 0 & 1 & 2000 \end{array} |) + 0 + 0 + 1500 | \begin{array}{c} 0 & 0 & 1 & 1 \\ 1 & 0 & 1 & 0 \\ 0 & 1 & 0 & 1 \\ 1 & 4 & 1 & 2 \end{array} |$

خطوة 5.4.4.2

بسّط المحدد.

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خطوة 5.4.4.2.1

بسّط كل حد.

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خطوة 5.4.4.2.1.1

اضرب $2000$ في $1$ .

$1 \cdot - 620 - 1 (0 - 1 (1 \cdot 4620 + 0 + 1 (2000 - 1 \cdot 1500)) + 0 - 1 | \begin{array}{c} 1 & 1 & 1500 \\ 1 & 0 & 1000 \\ 0 & 1 & 2000 \end{array} |) + 0 + 0 + 1500 | \begin{array}{c} 0 & 0 & 1 & 1 \\ 1 & 0 & 1 & 0 \\ 0 & 1 & 0 & 1 \\ 1 & 4 & 1 & 2 \end{array} |$

خطوة 5.4.4.2.1.2

اضرب $- 1$ في $1500$ .

$1 \cdot - 620 - 1 (0 - 1 (1 \cdot 4620 + 0 + 1 (2000 - 1500)) + 0 - 1 | \begin{array}{c} 1 & 1 & 1500 \\ 1 & 0 & 1000 \\ 0 & 1 & 2000 \end{array} |) + 0 + 0 + 1500 | \begin{array}{c} 0 & 0 & 1 & 1 \\ 1 & 0 & 1 & 0 \\ 0 & 1 & 0 & 1 \\ 1 & 4 & 1 & 2 \end{array} |$

خطوة 5.4.4.2.2

اطرح $1500$ من $2000$ .

$1 \cdot - 620 - 1 (0 - 1 (1 \cdot 4620 + 0 + 1 \cdot 500) + 0 - 1 | \begin{array}{c} 1 & 1 & 1500 \\ 1 & 0 & 1000 \\ 0 & 1 & 2000 \end{array} |) + 0 + 0 + 1500 | \begin{array}{c} 0 & 0 & 1 & 1 \\ 1 & 0 & 1 & 0 \\ 0 & 1 & 0 & 1 \\ 1 & 4 & 1 & 2 \end{array} |$

خطوة 5.4.5

بسّط المحدد.

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خطوة 5.4.5.1

بسّط كل حد.

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خطوة 5.4.5.1.1

اضرب $4620$ في $1$ .

$1 \cdot - 620 - 1 (0 - 1 (4620 + 0 + 1 \cdot 500) + 0 - 1 | \begin{array}{c} 1 & 1 & 1500 \\ 1 & 0 & 1000 \\ 0 & 1 & 2000 \end{array} |) + 0 + 0 + 1500 | \begin{array}{c} 0 & 0 & 1 & 1 \\ 1 & 0 & 1 & 0 \\ 0 & 1 & 0 & 1 \\ 1 & 4 & 1 & 2 \end{array} |$

خطوة 5.4.5.1.2

اضرب $500$ في $1$ .

$1 \cdot - 620 - 1 (0 - 1 (4620 + 0 + 500) + 0 - 1 | \begin{array}{c} 1 & 1 & 1500 \\ 1 & 0 & 1000 \\ 0 & 1 & 2000 \end{array} |) + 0 + 0 + 1500 | \begin{array}{c} 0 & 0 & 1 & 1 \\ 1 & 0 & 1 & 0 \\ 0 & 1 & 0 & 1 \\ 1 & 4 & 1 & 2 \end{array} |$

خطوة 5.4.5.2

أضف $4620$ و $0$ .

$1 \cdot - 620 - 1 (0 - 1 (4620 + 500) + 0 - 1 | \begin{array}{c} 1 & 1 & 1500 \\ 1 & 0 & 1000 \\ 0 & 1 & 2000 \end{array} |) + 0 + 0 + 1500 | \begin{array}{c} 0 & 0 & 1 & 1 \\ 1 & 0 & 1 & 0 \\ 0 & 1 & 0 & 1 \\ 1 & 4 & 1 & 2 \end{array} |$

خطوة 5.4.5.3

أضف $4620$ و $500$ .

$1 \cdot - 620 - 1 (0 - 1 \cdot 5120 + 0 - 1 | \begin{array}{c} 1 & 1 & 1500 \\ 1 & 0 & 1000 \\ 0 & 1 & 2000 \end{array} |) + 0 + 0 + 1500 | \begin{array}{c} 0 & 0 & 1 & 1 \\ 1 & 0 & 1 & 0 \\ 0 & 1 & 0 & 1 \\ 1 & 4 & 1 & 2 \end{array} |$

خطوة 5.5

احسِب قيمة $| \begin{array}{c} 1 & 1 & 1500 \\ 1 & 0 & 1000 \\ 0 & 1 & 2000 \end{array} |$ .

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خطوة 5.5.1

Choose the row or column with the most $0$ elements. If there are no $0$ elements choose any row or column. Multiply every element in column $1$ by its cofactor and add.

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خطوة 5.5.1.1

Consider the corresponding sign chart.

$| \begin{array}{c} + & - & + \\ - & + & - \\ + & - & + \end{array} |$

خطوة 5.5.1.2

The cofactor is the minor with the sign changed if the indices match a $-$ position on the sign chart.

خطوة 5.5.1.3

The minor for $a_{11}$ is the determinant with row $1$ and column $1$ deleted.

$| \begin{array}{c} 0 & 1000 \\ 1 & 2000 \end{array} |$

خطوة 5.5.1.4

Multiply element $a_{11}$ by its cofactor.

$1 | \begin{array}{c} 0 & 1000 \\ 1 & 2000 \end{array} |$

خطوة 5.5.1.5

The minor for $a_{21}$ is the determinant with row $2$ and column $1$ deleted.

$| \begin{array}{c} 1 & 1500 \\ 1 & 2000 \end{array} |$

خطوة 5.5.1.6

Multiply element $a_{21}$ by its cofactor.

$- 1 | \begin{array}{c} 1 & 1500 \\ 1 & 2000 \end{array} |$

خطوة 5.5.1.7

The minor for $a_{31}$ is the determinant with row $3$ and column $1$ deleted.

$| \begin{array}{c} 1 & 1500 \\ 0 & 1000 \end{array} |$

خطوة 5.5.1.8

Multiply element $a_{31}$ by its cofactor.

$0 | \begin{array}{c} 1 & 1500 \\ 0 & 1000 \end{array} |$

خطوة 5.5.1.9

Add the terms together.

$1 \cdot - 620 - 1 (0 - 1 \cdot 5120 + 0 - 1 (1 | \begin{array}{c} 0 & 1000 \\ 1 & 2000 \end{array} | - 1 | \begin{array}{c} 1 & 1500 \\ 1 & 2000 \end{array} | + 0 | \begin{array}{c} 1 & 1500 \\ 0 & 1000 \end{array} |)) + 0 + 0 + 1500 | \begin{array}{c} 0 & 0 & 1 & 1 \\ 1 & 0 & 1 & 0 \\ 0 & 1 & 0 & 1 \\ 1 & 4 & 1 & 2 \end{array} |$

خطوة 5.5.2

اضرب $0$ في $| \begin{array}{c} 1 & 1500 \\ 0 & 1000 \end{array} |$ .

$1 \cdot - 620 - 1 (0 - 1 \cdot 5120 + 0 - 1 (1 | \begin{array}{c} 0 & 1000 \\ 1 & 2000 \end{array} | - 1 | \begin{array}{c} 1 & 1500 \\ 1 & 2000 \end{array} | + 0)) + 0 + 0 + 1500 | \begin{array}{c} 0 & 0 & 1 & 1 \\ 1 & 0 & 1 & 0 \\ 0 & 1 & 0 & 1 \\ 1 & 4 & 1 & 2 \end{array} |$

خطوة 5.5.3

احسِب قيمة $| \begin{array}{c} 0 & 1000 \\ 1 & 2000 \end{array} |$ .

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خطوة 5.5.3.1

يمكن إيجاد محدد المصفوفة $2 \times 2$ باستخدام القاعدة $| \begin{array}{c} a & b \\ c & d \end{array} | = a d - c b$ .

$1 \cdot - 620 - 1 (0 - 1 \cdot 5120 + 0 - 1 (1 (0 \cdot 2000 - 1 \cdot 1000) - 1 | \begin{array}{c} 1 & 1500 \\ 1 & 2000 \end{array} | + 0)) + 0 + 0 + 1500 | \begin{array}{c} 0 & 0 & 1 & 1 \\ 1 & 0 & 1 & 0 \\ 0 & 1 & 0 & 1 \\ 1 & 4 & 1 & 2 \end{array} |$

خطوة 5.5.3.2

بسّط المحدد.

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خطوة 5.5.3.2.1

بسّط كل حد.

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خطوة 5.5.3.2.1.1

اضرب $0$ في $2000$ .

$1 \cdot - 620 - 1 (0 - 1 \cdot 5120 + 0 - 1 (1 (0 - 1 \cdot 1000) - 1 | \begin{array}{c} 1 & 1500 \\ 1 & 2000 \end{array} | + 0)) + 0 + 0 + 1500 | \begin{array}{c} 0 & 0 & 1 & 1 \\ 1 & 0 & 1 & 0 \\ 0 & 1 & 0 & 1 \\ 1 & 4 & 1 & 2 \end{array} |$

خطوة 5.5.3.2.1.2

اضرب $- 1$ في $1000$ .

$1 \cdot - 620 - 1 (0 - 1 \cdot 5120 + 0 - 1 (1 (0 - 1000) - 1 | \begin{array}{c} 1 & 1500 \\ 1 & 2000 \end{array} | + 0)) + 0 + 0 + 1500 | \begin{array}{c} 0 & 0 & 1 & 1 \\ 1 & 0 & 1 & 0 \\ 0 & 1 & 0 & 1 \\ 1 & 4 & 1 & 2 \end{array} |$

خطوة 5.5.3.2.2

اطرح $1000$ من $0$ .

$1 \cdot - 620 - 1 (0 - 1 \cdot 5120 + 0 - 1 (1 \cdot - 1000 - 1 | \begin{array}{c} 1 & 1500 \\ 1 & 2000 \end{array} | + 0)) + 0 + 0 + 1500 | \begin{array}{c} 0 & 0 & 1 & 1 \\ 1 & 0 & 1 & 0 \\ 0 & 1 & 0 & 1 \\ 1 & 4 & 1 & 2 \end{array} |$

خطوة 5.5.4

احسِب قيمة $| \begin{array}{c} 1 & 1500 \\ 1 & 2000 \end{array} |$ .

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خطوة 5.5.4.1

يمكن إيجاد محدد المصفوفة $2 \times 2$ باستخدام القاعدة $| \begin{array}{c} a & b \\ c & d \end{array} | = a d - c b$ .

$1 \cdot - 620 - 1 (0 - 1 \cdot 5120 + 0 - 1 (1 \cdot - 1000 - 1 (1 \cdot 2000 - 1 \cdot 1500) + 0)) + 0 + 0 + 1500 | \begin{array}{c} 0 & 0 & 1 & 1 \\ 1 & 0 & 1 & 0 \\ 0 & 1 & 0 & 1 \\ 1 & 4 & 1 & 2 \end{array} |$

خطوة 5.5.4.2

بسّط المحدد.

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خطوة 5.5.4.2.1

بسّط كل حد.

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خطوة 5.5.4.2.1.1

اضرب $2000$ في $1$ .

$1 \cdot - 620 - 1 (0 - 1 \cdot 5120 + 0 - 1 (1 \cdot - 1000 - 1 (2000 - 1 \cdot 1500) + 0)) + 0 + 0 + 1500 | \begin{array}{c} 0 & 0 & 1 & 1 \\ 1 & 0 & 1 & 0 \\ 0 & 1 & 0 & 1 \\ 1 & 4 & 1 & 2 \end{array} |$

خطوة 5.5.4.2.1.2

اضرب $- 1$ في $1500$ .

$1 \cdot - 620 - 1 (0 - 1 \cdot 5120 + 0 - 1 (1 \cdot - 1000 - 1 (2000 - 1500) + 0)) + 0 + 0 + 1500 | \begin{array}{c} 0 & 0 & 1 & 1 \\ 1 & 0 & 1 & 0 \\ 0 & 1 & 0 & 1 \\ 1 & 4 & 1 & 2 \end{array} |$

خطوة 5.5.4.2.2

اطرح $1500$ من $2000$ .

$1 \cdot - 620 - 1 (0 - 1 \cdot 5120 + 0 - 1 (1 \cdot - 1000 - 1 \cdot 500 + 0)) + 0 + 0 + 1500 | \begin{array}{c} 0 & 0 & 1 & 1 \\ 1 & 0 & 1 & 0 \\ 0 & 1 & 0 & 1 \\ 1 & 4 & 1 & 2 \end{array} |$

خطوة 5.5.5

بسّط المحدد.

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خطوة 5.5.5.1

بسّط كل حد.

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خطوة 5.5.5.1.1

اضرب $- 1000$ في $1$ .

$1 \cdot - 620 - 1 (0 - 1 \cdot 5120 + 0 - 1 (- 1000 - 1 \cdot 500 + 0)) + 0 + 0 + 1500 | \begin{array}{c} 0 & 0 & 1 & 1 \\ 1 & 0 & 1 & 0 \\ 0 & 1 & 0 & 1 \\ 1 & 4 & 1 & 2 \end{array} |$

خطوة 5.5.5.1.2

اضرب $- 1$ في $500$ .

$1 \cdot - 620 - 1 (0 - 1 \cdot 5120 + 0 - 1 (- 1000 - 500 + 0)) + 0 + 0 + 1500 | \begin{array}{c} 0 & 0 & 1 & 1 \\ 1 & 0 & 1 & 0 \\ 0 & 1 & 0 & 1 \\ 1 & 4 & 1 & 2 \end{array} |$

خطوة 5.5.5.2

اطرح $500$ من $- 1000$ .

$1 \cdot - 620 - 1 (0 - 1 \cdot 5120 + 0 - 1 (- 1500 + 0)) + 0 + 0 + 1500 | \begin{array}{c} 0 & 0 & 1 & 1 \\ 1 & 0 & 1 & 0 \\ 0 & 1 & 0 & 1 \\ 1 & 4 & 1 & 2 \end{array} |$

خطوة 5.5.5.3

أضف $- 1500$ و $0$ .

$1 \cdot - 620 - 1 (0 - 1 \cdot 5120 + 0 - 1 \cdot - 1500) + 0 + 0 + 1500 | \begin{array}{c} 0 & 0 & 1 & 1 \\ 1 & 0 & 1 & 0 \\ 0 & 1 & 0 & 1 \\ 1 & 4 & 1 & 2 \end{array} |$

خطوة 5.6

بسّط المحدد.

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خطوة 5.6.1

بسّط كل حد.

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خطوة 5.6.1.1

اضرب $- 1$ في $5120$ .

$1 \cdot - 620 - 1 (0 - 5120 + 0 - 1 \cdot - 1500) + 0 + 0 + 1500 | \begin{array}{c} 0 & 0 & 1 & 1 \\ 1 & 0 & 1 & 0 \\ 0 & 1 & 0 & 1 \\ 1 & 4 & 1 & 2 \end{array} |$

خطوة 5.6.1.2

اضرب $- 1$ في $- 1500$ .

$1 \cdot - 620 - 1 (0 - 5120 + 0 + 1500) + 0 + 0 + 1500 | \begin{array}{c} 0 & 0 & 1 & 1 \\ 1 & 0 & 1 & 0 \\ 0 & 1 & 0 & 1 \\ 1 & 4 & 1 & 2 \end{array} |$

خطوة 5.6.2

اطرح $5120$ من $0$ .

$1 \cdot - 620 - 1 (- 5120 + 0 + 1500) + 0 + 0 + 1500 | \begin{array}{c} 0 & 0 & 1 & 1 \\ 1 & 0 & 1 & 0 \\ 0 & 1 & 0 & 1 \\ 1 & 4 & 1 & 2 \end{array} |$

خطوة 5.6.3

أضف $- 5120$ و $0$ .

$1 \cdot - 620 - 1 (- 5120 + 1500) + 0 + 0 + 1500 | \begin{array}{c} 0 & 0 & 1 & 1 \\ 1 & 0 & 1 & 0 \\ 0 & 1 & 0 & 1 \\ 1 & 4 & 1 & 2 \end{array} |$

خطوة 5.6.4

أضف $- 5120$ و $1500$ .

$1 \cdot - 620 - 1 \cdot - 3620 + 0 + 0 + 1500 | \begin{array}{c} 0 & 0 & 1 & 1 \\ 1 & 0 & 1 & 0 \\ 0 & 1 & 0 & 1 \\ 1 & 4 & 1 & 2 \end{array} |$

خطوة 6

احسِب قيمة $| \begin{array}{c} 0 & 0 & 1 & 1 \\ 1 & 0 & 1 & 0 \\ 0 & 1 & 0 & 1 \\ 1 & 4 & 1 & 2 \end{array} |$ .

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خطوة 6.1

Choose the row or column with the most $0$ elements. If there are no $0$ elements choose any row or column. Multiply every element in row $1$ by its cofactor and add.

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خطوة 6.1.1

Consider the corresponding sign chart.

$| \begin{array}{c} + & - & + & - \\ - & + & - & + \\ + & - & + & - \\ - & + & - & + \end{array} |$

خطوة 6.1.2

The cofactor is the minor with the sign changed if the indices match a $-$ position on the sign chart.

خطوة 6.1.3

The minor for $a_{11}$ is the determinant with row $1$ and column $1$ deleted.

$| \begin{array}{c} 0 & 1 & 0 \\ 1 & 0 & 1 \\ 4 & 1 & 2 \end{array} |$

خطوة 6.1.4

Multiply element $a_{11}$ by its cofactor.

$0 | \begin{array}{c} 0 & 1 & 0 \\ 1 & 0 & 1 \\ 4 & 1 & 2 \end{array} |$

خطوة 6.1.5

The minor for $a_{12}$ is the determinant with row $1$ and column $2$ deleted.

$| \begin{array}{c} 1 & 1 & 0 \\ 0 & 0 & 1 \\ 1 & 1 & 2 \end{array} |$

خطوة 6.1.6

Multiply element $a_{12}$ by its cofactor.

$0 | \begin{array}{c} 1 & 1 & 0 \\ 0 & 0 & 1 \\ 1 & 1 & 2 \end{array} |$

خطوة 6.1.7

The minor for $a_{13}$ is the determinant with row $1$ and column $3$ deleted.

$| \begin{array}{c} 1 & 0 & 0 \\ 0 & 1 & 1 \\ 1 & 4 & 2 \end{array} |$

خطوة 6.1.8

Multiply element $a_{13}$ by its cofactor.

$1 | \begin{array}{c} 1 & 0 & 0 \\ 0 & 1 & 1 \\ 1 & 4 & 2 \end{array} |$

خطوة 6.1.9

The minor for $a_{14}$ is the determinant with row $1$ and column $4$ deleted.

$| \begin{array}{c} 1 & 0 & 1 \\ 0 & 1 & 0 \\ 1 & 4 & 1 \end{array} |$

خطوة 6.1.10

Multiply element $a_{14}$ by its cofactor.

$- 1 | \begin{array}{c} 1 & 0 & 1 \\ 0 & 1 & 0 \\ 1 & 4 & 1 \end{array} |$

خطوة 6.1.11

Add the terms together.

$1 \cdot - 620 - 1 \cdot - 3620 + 0 + 0 + 1500 (0 | \begin{array}{c} 0 & 1 & 0 \\ 1 & 0 & 1 \\ 4 & 1 & 2 \end{array} | + 0 | \begin{array}{c} 1 & 1 & 0 \\ 0 & 0 & 1 \\ 1 & 1 & 2 \end{array} | + 1 | \begin{array}{c} 1 & 0 & 0 \\ 0 & 1 & 1 \\ 1 & 4 & 2 \end{array} | - 1 | \begin{array}{c} 1 & 0 & 1 \\ 0 & 1 & 0 \\ 1 & 4 & 1 \end{array} |)$

خطوة 6.2

اضرب $0$ في $| \begin{array}{c} 0 & 1 & 0 \\ 1 & 0 & 1 \\ 4 & 1 & 2 \end{array} |$ .

$1 \cdot - 620 - 1 \cdot - 3620 + 0 + 0 + 1500 (0 + 0 | \begin{array}{c} 1 & 1 & 0 \\ 0 & 0 & 1 \\ 1 & 1 & 2 \end{array} | + 1 | \begin{array}{c} 1 & 0 & 0 \\ 0 & 1 & 1 \\ 1 & 4 & 2 \end{array} | - 1 | \begin{array}{c} 1 & 0 & 1 \\ 0 & 1 & 0 \\ 1 & 4 & 1 \end{array} |)$

خطوة 6.3

اضرب $0$ في $| \begin{array}{c} 1 & 1 & 0 \\ 0 & 0 & 1 \\ 1 & 1 & 2 \end{array} |$ .

$1 \cdot - 620 - 1 \cdot - 3620 + 0 + 0 + 1500 (0 + 0 + 1 | \begin{array}{c} 1 & 0 & 0 \\ 0 & 1 & 1 \\ 1 & 4 & 2 \end{array} | - 1 | \begin{array}{c} 1 & 0 & 1 \\ 0 & 1 & 0 \\ 1 & 4 & 1 \end{array} |)$

خطوة 6.4

احسِب قيمة $| \begin{array}{c} 1 & 0 & 0 \\ 0 & 1 & 1 \\ 1 & 4 & 2 \end{array} |$ .

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خطوة 6.4.1

Choose the row or column with the most $0$ elements. If there are no $0$ elements choose any row or column. Multiply every element in row $1$ by its cofactor and add.

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خطوة 6.4.1.1

Consider the corresponding sign chart.

$| \begin{array}{c} + & - & + \\ - & + & - \\ + & - & + \end{array} |$

خطوة 6.4.1.2

The cofactor is the minor with the sign changed if the indices match a $-$ position on the sign chart.

خطوة 6.4.1.3

The minor for $a_{11}$ is the determinant with row $1$ and column $1$ deleted.

$| \begin{array}{c} 1 & 1 \\ 4 & 2 \end{array} |$

خطوة 6.4.1.4

Multiply element $a_{11}$ by its cofactor.

$1 | \begin{array}{c} 1 & 1 \\ 4 & 2 \end{array} |$

خطوة 6.4.1.5

The minor for $a_{12}$ is the determinant with row $1$ and column $2$ deleted.

$| \begin{array}{c} 0 & 1 \\ 1 & 2 \end{array} |$

خطوة 6.4.1.6

Multiply element $a_{12}$ by its cofactor.

$0 | \begin{array}{c} 0 & 1 \\ 1 & 2 \end{array} |$

خطوة 6.4.1.7

The minor for $a_{13}$ is the determinant with row $1$ and column $3$ deleted.

$| \begin{array}{c} 0 & 1 \\ 1 & 4 \end{array} |$

خطوة 6.4.1.8

Multiply element $a_{13}$ by its cofactor.

$0 | \begin{array}{c} 0 & 1 \\ 1 & 4 \end{array} |$

خطوة 6.4.1.9

Add the terms together.

$1 \cdot - 620 - 1 \cdot - 3620 + 0 + 0 + 1500 (0 + 0 + 1 (1 | \begin{array}{c} 1 & 1 \\ 4 & 2 \end{array} | + 0 | \begin{array}{c} 0 & 1 \\ 1 & 2 \end{array} | + 0 | \begin{array}{c} 0 & 1 \\ 1 & 4 \end{array} |) - 1 | \begin{array}{c} 1 & 0 & 1 \\ 0 & 1 & 0 \\ 1 & 4 & 1 \end{array} |)$

خطوة 6.4.2

اضرب $0$ في $| \begin{array}{c} 0 & 1 \\ 1 & 2 \end{array} |$ .

$1 \cdot - 620 - 1 \cdot - 3620 + 0 + 0 + 1500 (0 + 0 + 1 (1 | \begin{array}{c} 1 & 1 \\ 4 & 2 \end{array} | + 0 + 0 | \begin{array}{c} 0 & 1 \\ 1 & 4 \end{array} |) - 1 | \begin{array}{c} 1 & 0 & 1 \\ 0 & 1 & 0 \\ 1 & 4 & 1 \end{array} |)$

خطوة 6.4.3

اضرب $0$ في $| \begin{array}{c} 0 & 1 \\ 1 & 4 \end{array} |$ .

$1 \cdot - 620 - 1 \cdot - 3620 + 0 + 0 + 1500 (0 + 0 + 1 (1 | \begin{array}{c} 1 & 1 \\ 4 & 2 \end{array} | + 0 + 0) - 1 | \begin{array}{c} 1 & 0 & 1 \\ 0 & 1 & 0 \\ 1 & 4 & 1 \end{array} |)$

خطوة 6.4.4

احسِب قيمة $| \begin{array}{c} 1 & 1 \\ 4 & 2 \end{array} |$ .

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خطوة 6.4.4.1

يمكن إيجاد محدد المصفوفة $2 \times 2$ باستخدام القاعدة $| \begin{array}{c} a & b \\ c & d \end{array} | = a d - c b$ .

$1 \cdot - 620 - 1 \cdot - 3620 + 0 + 0 + 1500 (0 + 0 + 1 (1 (1 \cdot 2 - 4 \cdot 1) + 0 + 0) - 1 | \begin{array}{c} 1 & 0 & 1 \\ 0 & 1 & 0 \\ 1 & 4 & 1 \end{array} |)$

خطوة 6.4.4.2

بسّط المحدد.

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خطوة 6.4.4.2.1

بسّط كل حد.

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خطوة 6.4.4.2.1.1

اضرب $2$ في $1$ .

$1 \cdot - 620 - 1 \cdot - 3620 + 0 + 0 + 1500 (0 + 0 + 1 (1 (2 - 4 \cdot 1) + 0 + 0) - 1 | \begin{array}{c} 1 & 0 & 1 \\ 0 & 1 & 0 \\ 1 & 4 & 1 \end{array} |)$

خطوة 6.4.4.2.1.2

اضرب $- 4$ في $1$ .

$1 \cdot - 620 - 1 \cdot - 3620 + 0 + 0 + 1500 (0 + 0 + 1 (1 (2 - 4) + 0 + 0) - 1 | \begin{array}{c} 1 & 0 & 1 \\ 0 & 1 & 0 \\ 1 & 4 & 1 \end{array} |)$

خطوة 6.4.4.2.2

اطرح $4$ من $2$ .

$1 \cdot - 620 - 1 \cdot - 3620 + 0 + 0 + 1500 (0 + 0 + 1 (1 \cdot - 2 + 0 + 0) - 1 | \begin{array}{c} 1 & 0 & 1 \\ 0 & 1 & 0 \\ 1 & 4 & 1 \end{array} |)$

خطوة 6.4.5

بسّط المحدد.

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خطوة 6.4.5.1

اضرب $- 2$ في $1$ .

$1 \cdot - 620 - 1 \cdot - 3620 + 0 + 0 + 1500 (0 + 0 + 1 (- 2 + 0 + 0) - 1 | \begin{array}{c} 1 & 0 & 1 \\ 0 & 1 & 0 \\ 1 & 4 & 1 \end{array} |)$

خطوة 6.4.5.2

أضف $- 2$ و $0$ .

$1 \cdot - 620 - 1 \cdot - 3620 + 0 + 0 + 1500 (0 + 0 + 1 (- 2 + 0) - 1 | \begin{array}{c} 1 & 0 & 1 \\ 0 & 1 & 0 \\ 1 & 4 & 1 \end{array} |)$

خطوة 6.4.5.3

أضف $- 2$ و $0$ .

$1 \cdot - 620 - 1 \cdot - 3620 + 0 + 0 + 1500 (0 + 0 + 1 \cdot - 2 - 1 | \begin{array}{c} 1 & 0 & 1 \\ 0 & 1 & 0 \\ 1 & 4 & 1 \end{array} |)$

خطوة 6.5

احسِب قيمة $| \begin{array}{c} 1 & 0 & 1 \\ 0 & 1 & 0 \\ 1 & 4 & 1 \end{array} |$ .

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خطوة 6.5.1

Choose the row or column with the most $0$ elements. If there are no $0$ elements choose any row or column. Multiply every element in row $2$ by its cofactor and add.

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خطوة 6.5.1.1

Consider the corresponding sign chart.

$| \begin{array}{c} + & - & + \\ - & + & - \\ + & - & + \end{array} |$

خطوة 6.5.1.2

The cofactor is the minor with the sign changed if the indices match a $-$ position on the sign chart.

خطوة 6.5.1.3

The minor for $a_{21}$ is the determinant with row $2$ and column $1$ deleted.

$| \begin{array}{c} 0 & 1 \\ 4 & 1 \end{array} |$

خطوة 6.5.1.4

Multiply element $a_{21}$ by its cofactor.

$0 | \begin{array}{c} 0 & 1 \\ 4 & 1 \end{array} |$

خطوة 6.5.1.5

The minor for $a_{22}$ is the determinant with row $2$ and column $2$ deleted.

$| \begin{array}{c} 1 & 1 \\ 1 & 1 \end{array} |$

خطوة 6.5.1.6

Multiply element $a_{22}$ by its cofactor.

$1 | \begin{array}{c} 1 & 1 \\ 1 & 1 \end{array} |$

خطوة 6.5.1.7

The minor for $a_{23}$ is the determinant with row $2$ and column $3$ deleted.

$| \begin{array}{c} 1 & 0 \\ 1 & 4 \end{array} |$

خطوة 6.5.1.8

Multiply element $a_{23}$ by its cofactor.

$0 | \begin{array}{c} 1 & 0 \\ 1 & 4 \end{array} |$

خطوة 6.5.1.9

Add the terms together.

$1 \cdot - 620 - 1 \cdot - 3620 + 0 + 0 + 1500 (0 + 0 + 1 \cdot - 2 - 1 (0 | \begin{array}{c} 0 & 1 \\ 4 & 1 \end{array} | + 1 | \begin{array}{c} 1 & 1 \\ 1 & 1 \end{array} | + 0 | \begin{array}{c} 1 & 0 \\ 1 & 4 \end{array} |))$

خطوة 6.5.2

اضرب $0$ في $| \begin{array}{c} 0 & 1 \\ 4 & 1 \end{array} |$ .

$1 \cdot - 620 - 1 \cdot - 3620 + 0 + 0 + 1500 (0 + 0 + 1 \cdot - 2 - 1 (0 + 1 | \begin{array}{c} 1 & 1 \\ 1 & 1 \end{array} | + 0 | \begin{array}{c} 1 & 0 \\ 1 & 4 \end{array} |))$

خطوة 6.5.3

اضرب $0$ في $| \begin{array}{c} 1 & 0 \\ 1 & 4 \end{array} |$ .

$1 \cdot - 620 - 1 \cdot - 3620 + 0 + 0 + 1500 (0 + 0 + 1 \cdot - 2 - 1 (0 + 1 | \begin{array}{c} 1 & 1 \\ 1 & 1 \end{array} | + 0))$

خطوة 6.5.4

احسِب قيمة $| \begin{array}{c} 1 & 1 \\ 1 & 1 \end{array} |$ .

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خطوة 6.5.4.1

يمكن إيجاد محدد المصفوفة $2 \times 2$ باستخدام القاعدة $| \begin{array}{c} a & b \\ c & d \end{array} | = a d - c b$ .

$1 \cdot - 620 - 1 \cdot - 3620 + 0 + 0 + 1500 (0 + 0 + 1 \cdot - 2 - 1 (0 + 1 (1 \cdot 1 - 1 \cdot 1) + 0))$

خطوة 6.5.4.2

بسّط المحدد.

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خطوة 6.5.4.2.1

بسّط كل حد.

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خطوة 6.5.4.2.1.1

اضرب $1$ في $1$ .

$1 \cdot - 620 - 1 \cdot - 3620 + 0 + 0 + 1500 (0 + 0 + 1 \cdot - 2 - 1 (0 + 1 (1 - 1 \cdot 1) + 0))$

خطوة 6.5.4.2.1.2

اضرب $- 1$ في $1$ .

$1 \cdot - 620 - 1 \cdot - 3620 + 0 + 0 + 1500 (0 + 0 + 1 \cdot - 2 - 1 (0 + 1 (1 - 1) + 0))$

خطوة 6.5.4.2.2

اطرح $1$ من $1$ .

$1 \cdot - 620 - 1 \cdot - 3620 + 0 + 0 + 1500 (0 + 0 + 1 \cdot - 2 - 1 (0 + 1 \cdot 0 + 0))$

خطوة 6.5.5

بسّط المحدد.

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خطوة 6.5.5.1

اضرب $0$ في $1$ .

$1 \cdot - 620 - 1 \cdot - 3620 + 0 + 0 + 1500 (0 + 0 + 1 \cdot - 2 - 1 (0 + 0 + 0))$

خطوة 6.5.5.2

أضف $0$ و $0$ .

$1 \cdot - 620 - 1 \cdot - 3620 + 0 + 0 + 1500 (0 + 0 + 1 \cdot - 2 - 1 (0 + 0))$

خطوة 6.5.5.3

أضف $0$ و $0$ .

$1 \cdot - 620 - 1 \cdot - 3620 + 0 + 0 + 1500 (0 + 0 + 1 \cdot - 2 - 1 \cdot 0)$

خطوة 6.6

بسّط المحدد.

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خطوة 6.6.1

اضرب $- 2$ في $1$ .

$1 \cdot - 620 - 1 \cdot - 3620 + 0 + 0 + 1500 (0 + 0 - 2 - 1 \cdot 0)$

خطوة 6.6.2

أضف $0$ و $0$ .

$1 \cdot - 620 - 1 \cdot - 3620 + 0 + 0 + 1500 (0 - 2 - 1 \cdot 0)$

خطوة 6.6.3

اطرح $2$ من $0$ .

$1 \cdot - 620 - 1 \cdot - 3620 + 0 + 0 + 1500 (- 2 - 1 \cdot 0)$

خطوة 6.6.4

اطرح $0$ من $- 2$ .

$1 \cdot - 620 - 1 \cdot - 3620 + 0 + 0 + 1500 \cdot - 2$

خطوة 7

بسّط المحدد.

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خطوة 7.1

بسّط كل حد.

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خطوة 7.1.1

اضرب $- 620$ في $1$ .

$- 620 - 1 \cdot - 3620 + 0 + 0 + 1500 \cdot - 2$

خطوة 7.1.2

اضرب $- 1$ في $- 3620$ .

$- 620 + 3620 + 0 + 0 + 1500 \cdot - 2$

خطوة 7.1.3

اضرب $1500$ في $- 2$ .

$- 620 + 3620 + 0 + 0 - 3000$

خطوة 7.2

أضف $- 620$ و $3620$ .

$3000 + 0 + 0 - 3000$

خطوة 7.3

أضف $3000$ و $0$ .

$3000 + 0 - 3000$

خطوة 7.4

أضف $3000$ و $0$ .

$3000 - 3000$

خطوة 7.5

اطرح $3000$ من $3000$ .

$0$