الجبر الخطي الأمثلة

[\begin{array}{c} 1 & 7 & 1 \\ 2 & 9 & 0 \\ 1 & 8 & 1 \end{array}] [\begin{array}{c} 4.5 & 0.5 & - 4.5 \\ - 1 & 0 & 1 \\ 3.5 & - 0.5 & - 2.5 \end{array}]

$[\begin{array}{c} 1 & 7 & 1 \\ 2 & 9 & 0 \\ 1 & 8 & 1 \end{array}] [\begin{array}{c} 4.5 & 0.5 & - 4.5 \\ - 1 & 0 & 1 \\ 3.5 & - 0.5 & - 2.5 \end{array}]$

خطوة 1

اضرب

[\begin{array}{c} 1 & 7 & 1 \\ 2 & 9 & 0 \\ 1 & 8 & 1 \end{array}] [\begin{array}{c} 4.5 & 0.5 & - 4.5 \\ - 1 & 0 & 1 \\ 3.5 & - 0.5 & - 2.5 \end{array}]

$[\begin{array}{c} 1 & 7 & 1 \\ 2 & 9 & 0 \\ 1 & 8 & 1 \end{array}] [\begin{array}{c} 4.5 & 0.5 & - 4.5 \\ - 1 & 0 & 1 \\ 3.5 & - 0.5 & - 2.5 \end{array}]$ .

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خطوة 1.1

Two matrices can be multiplied if and only if the number of columns in the first matrix is equal to the number of rows in the second matrix. In this case, the first matrix is

3 \times 3

$3 \times 3$ and the second matrix is

3 \times 3

$3 \times 3$ .

خطوة 1.2

اضرب كل صف في المصفوفة الأولى في كل عمود في المصفوفة الثانية.

[\begin{array}{c} 1 \cdot 4.5 + 7 \cdot - 1 + 1 \cdot 3.5 & 1 \cdot 0.5 + 7 \cdot 0 + 1 \cdot - 0.5 & 1 \cdot - 4.5 + 7 \cdot 1 + 1 \cdot - 2.5 \\ 2 \cdot 4.5 + 9 \cdot - 1 + 0 \cdot 3.5 & 2 \cdot 0.5 + 9 \cdot 0 + 0 \cdot - 0.5 & 2 \cdot - 4.5 + 9 \cdot 1 + 0 \cdot - 2.5 \\ 1 \cdot 4.5 + 8 \cdot - 1 + 1 \cdot 3.5 & 1 \cdot 0.5 + 8 \cdot 0 + 1 \cdot - 0.5 & 1 \cdot - 4.5 + 8 \cdot 1 + 1 \cdot - 2.5 \end{array}]

$[\begin{array}{c} 1 \cdot 4.5 + 7 \cdot - 1 + 1 \cdot 3.5 & 1 \cdot 0.5 + 7 \cdot 0 + 1 \cdot - 0.5 & 1 \cdot - 4.5 + 7 \cdot 1 + 1 \cdot - 2.5 \\ 2 \cdot 4.5 + 9 \cdot - 1 + 0 \cdot 3.5 & 2 \cdot 0.5 + 9 \cdot 0 + 0 \cdot - 0.5 & 2 \cdot - 4.5 + 9 \cdot 1 + 0 \cdot - 2.5 \\ 1 \cdot 4.5 + 8 \cdot - 1 + 1 \cdot 3.5 & 1 \cdot 0.5 + 8 \cdot 0 + 1 \cdot - 0.5 & 1 \cdot - 4.5 + 8 \cdot 1 + 1 \cdot - 2.5 \end{array}]$

خطوة 1.3

بسّط كل عنصر من عناصر المصفوفة بضرب جميع العبارات.

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

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

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

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

خطوة 2

Find the determinant.

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خطوة 2.1

Choose the row or column with the most

0

$0$ elements. If there are no

0

$0$ elements choose any row or column. Multiply every element in row

1

$1$ by its cofactor and add.

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خطوة 2.1.1

Consider the corresponding sign chart.

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

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

خطوة 2.1.2

The cofactor is the minor with the sign changed if the indices match a

-

$-$ position on the sign chart.

خطوة 2.1.3

The minor for

a_{11}

$a_{11}$ is the determinant with row

1

$1$ and column

1

$1$ deleted.

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

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

خطوة 2.1.4

Multiply element

a_{11}

$a_{11}$ by its cofactor.

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

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

خطوة 2.1.5

The minor for

a_{12}

$a_{12}$ is the determinant with row

1

$1$ and column

2

$2$ deleted.

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

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

خطوة 2.1.6

Multiply element

a_{12}

$a_{12}$ by its cofactor.

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

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

خطوة 2.1.7

The minor for

a_{13}

$a_{13}$ is the determinant with row

1

$1$ and column

3

$3$ deleted.

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

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

خطوة 2.1.8

Multiply element

a_{13}

$a_{13}$ by its cofactor.

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

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

خطوة 2.1.9

Add the terms together.

1 | \begin{array}{c} 1 & 0 \\ 0 & 1 \end{array} | + 0 | \begin{array}{c} 0 & 0 \\ 0 & 1 \end{array} | + 0 | \begin{array}{c} 0 & 1 \\ 0 & 0 \end{array} |

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

1 | \begin{array}{c} 1 & 0 \\ 0 & 1 \end{array} | + 0 | \begin{array}{c} 0 & 0 \\ 0 & 1 \end{array} | + 0 | \begin{array}{c} 0 & 1 \\ 0 & 0 \end{array} |

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

خطوة 2.2

اضرب

0

$0$ في

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

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

1 | \begin{array}{c} 1 & 0 \\ 0 & 1 \end{array} | + 0 + 0 | \begin{array}{c} 0 & 1 \\ 0 & 0 \end{array} |

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

خطوة 2.3

اضرب

0

$0$ في

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

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

1 | \begin{array}{c} 1 & 0 \\ 0 & 1 \end{array} | + 0 + 0

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

خطوة 2.4

احسِب قيمة

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

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

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خطوة 2.4.1

يمكن إيجاد محدد المصفوفة

2 \times 2

$2 \times 2$ باستخدام القاعدة

| \begin{array}{c} a & b \\ c & d \end{array} | = a d - c b

$| \begin{array}{c} a & b \\ c & d \end{array} | = a d - c b$ .

1 (1 \cdot 1 + 0 \cdot 0) + 0 + 0

$1 (1 \cdot 1 + 0 \cdot 0) + 0 + 0$

خطوة 2.4.2

بسّط المحدد.

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خطوة 2.4.2.1

بسّط كل حد.

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خطوة 2.4.2.1.1

اضرب

1

$1$ في

1

$1$ .

1 (1 + 0 \cdot 0) + 0 + 0

$1 (1 + 0 \cdot 0) + 0 + 0$

خطوة 2.4.2.1.2

اضرب

0

$0$ في

0

$0$ .

1 (1 + 0) + 0 + 0

$1 (1 + 0) + 0 + 0$

1 (1 + 0) + 0 + 0

$1 (1 + 0) + 0 + 0$

خطوة 2.4.2.2

أضف

1

$1$ و

0

$0$ .

1 \cdot 1 + 0 + 0

$1 \cdot 1 + 0 + 0$

1 \cdot 1 + 0 + 0

$1 \cdot 1 + 0 + 0$

1 \cdot 1 + 0 + 0

$1 \cdot 1 + 0 + 0$

خطوة 2.5

بسّط المحدد.

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خطوة 2.5.1

اضرب

1

$1$ في

1

$1$ .

1 + 0 + 0

$1 + 0 + 0$

خطوة 2.5.2

أضف

1

$1$ و

0

$0$ .

1 + 0

$1 + 0$

خطوة 2.5.3

أضف

1

$1$ و

0

$0$ .

1

$1$

1

$1$

1

$1$

خطوة 3

Since the determinant is non-zero, the inverse exists.

خطوة 4

Set up a

3 \times 6

$3 \times 6$ matrix where the left half is the original matrix and the right half is its identity matrix.

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

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

خطوة 5

The right half of the reduced row echelon form is the inverse.

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

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