الجبر الخطي الأمثلة

⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ \begin{matrix} 1 & 1 & 0 1 & 0 & 1 1 & 0 & 1 2 & 1 & 0 2 & 1 & 0 \end{matrix} ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

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

خطوة 1

Perform the row operation

$R_{2} = R_{2} - R_{1}$ to make the entry at

$2, 1$ a

$0$ .

انقر لعرض المزيد من الخطوات...

خطوة 1.1

Perform the row operation

$R_{2} = R_{2} - R_{1}$ to make the entry at

$2, 1$ a

$0$ .

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

خطوة 1.2

بسّط

$R_{2}$ .

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

خطوة 2

Perform the row operation

$R_{3} = R_{3} - R_{1}$ to make the entry at

$3, 1$ a

$0$ .

انقر لعرض المزيد من الخطوات...

خطوة 2.1

Perform the row operation

$R_{3} = R_{3} - R_{1}$ to make the entry at

$3, 1$ a

$0$ .

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

خطوة 2.2

بسّط

$R_{3}$ .

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

خطوة 3

Perform the row operation

$R_{4} = R_{4} - 2 R_{1}$ to make the entry at

$4, 1$ a

$0$ .

انقر لعرض المزيد من الخطوات...

خطوة 3.1

Perform the row operation

$R_{4} = R_{4} - 2 R_{1}$ to make the entry at

$4, 1$ a

$0$ .

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

خطوة 3.2

بسّط

$R_{4}$ .

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

خطوة 4

Perform the row operation

$R_{5} = R_{5} - 2 R_{1}$ to make the entry at

$5, 1$ a

$0$ .

انقر لعرض المزيد من الخطوات...

خطوة 4.1

Perform the row operation

$R_{5} = R_{5} - 2 R_{1}$ to make the entry at

$5, 1$ a

$0$ .

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

خطوة 4.2

بسّط

$R_{5}$ .

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

خطوة 5

Multiply each element of

$R_{2}$ by

$- 1$ to make the entry at

$2, 2$ a

$1$ .

انقر لعرض المزيد من الخطوات...

خطوة 5.1

Multiply each element of

$R_{2}$ by

$- 1$ to make the entry at

$2, 2$ a

$1$ .

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

خطوة 5.2

بسّط

$R_{2}$ .

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

خطوة 6

Perform the row operation

$R_{3} = R_{3} + R_{2}$ to make the entry at

$3, 2$ a

$0$ .

انقر لعرض المزيد من الخطوات...

خطوة 6.1

Perform the row operation

$R_{3} = R_{3} + R_{2}$ to make the entry at

$3, 2$ a

$0$ .

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

خطوة 6.2

بسّط

$R_{3}$ .

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

خطوة 7

Perform the row operation

$R_{4} = R_{4} + R_{2}$ to make the entry at

$4, 2$ a

$0$ .

انقر لعرض المزيد من الخطوات...

خطوة 7.1

Perform the row operation

$R_{4} = R_{4} + R_{2}$ to make the entry at

$4, 2$ a

$0$ .

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

خطوة 7.2

بسّط

$R_{4}$ .

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

خطوة 8

Perform the row operation

$R_{5} = R_{5} + R_{2}$ to make the entry at

$5, 2$ a

$0$ .

انقر لعرض المزيد من الخطوات...

خطوة 8.1

Perform the row operation

$R_{5} = R_{5} + R_{2}$ to make the entry at

$5, 2$ a

$0$ .

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

خطوة 8.2

بسّط

$R_{5}$ .

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

خطوة 9

Swap

$R_{4}$ with

$R_{3}$ to put a nonzero entry at

$3, 3$ .

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

خطوة 10

Multiply each element of

$R_{3}$ by

$- 1$ to make the entry at

$3, 3$ a

$1$ .

انقر لعرض المزيد من الخطوات...

خطوة 10.1

Multiply each element of

$R_{3}$ by

$- 1$ to make the entry at

$3, 3$ a

$1$ .

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

خطوة 10.2

بسّط

$R_{3}$ .

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

خطوة 11

Perform the row operation

$R_{5} = R_{5} + R_{3}$ to make the entry at

$5, 3$ a

$0$ .

انقر لعرض المزيد من الخطوات...

خطوة 11.1

Perform the row operation

$R_{5} = R_{5} + R_{3}$ to make the entry at

$5, 3$ a

$0$ .

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

خطوة 11.2

بسّط

$R_{5}$ .

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

خطوة 12

Perform the row operation

$R_{2} = R_{2} + R_{3}$ to make the entry at

$2, 3$ a

$0$ .

انقر لعرض المزيد من الخطوات...

خطوة 12.1

Perform the row operation

$R_{2} = R_{2} + R_{3}$ to make the entry at

$2, 3$ a

$0$ .

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

خطوة 12.2

بسّط

$R_{2}$ .

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

خطوة 13

Perform the row operation

$R_{1} = R_{1} - R_{2}$ to make the entry at

$1, 2$ a

$0$ .

انقر لعرض المزيد من الخطوات...

خطوة 13.1

Perform the row operation

$R_{1} = R_{1} - R_{2}$ to make the entry at

$1, 2$ a

$0$ .

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

خطوة 13.2

بسّط

$R_{1}$ .

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