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الجبر الخطي الأمثلة
[762631556]⎡⎢⎣762631556⎤⎥⎦
خطوة 1
Write as an augmented matrix for Ax=0Ax=0.
[762063105560]⎡⎢
⎢⎣762063105560⎤⎥
⎥⎦
خطوة 2
خطوة 2.1
Multiply each element of R1R1 by 1717 to make the entry at 1,11,1 a 11.
خطوة 2.1.1
Multiply each element of R1R1 by 1717 to make the entry at 1,11,1 a 11.
[7767270763105560]⎡⎢
⎢⎣7767270763105560⎤⎥
⎥⎦
خطوة 2.1.2
بسّط R1R1.
[16727063105560]⎡⎢
⎢⎣16727063105560⎤⎥
⎥⎦
[16727063105560]⎡⎢
⎢⎣16727063105560⎤⎥
⎥⎦
خطوة 2.2
Perform the row operation R2=R2-6R1R2=R2−6R1 to make the entry at 2,12,1 a 00.
خطوة 2.2.1
Perform the row operation R2=R2-6R1R2=R2−6R1 to make the entry at 2,12,1 a 00.
[1672706-6⋅13-6(67)1-6(27)0-6⋅05560]⎡⎢
⎢
⎢
⎢⎣1672706−6⋅13−6(67)1−6(27)0−6⋅05560⎤⎥
⎥
⎥
⎥⎦
خطوة 2.2.2
بسّط R2R2.
[1672700-157-5705560]⎡⎢
⎢
⎢⎣1672700−157−5705560⎤⎥
⎥
⎥⎦
[1672700-157-5705560]⎡⎢
⎢
⎢⎣1672700−157−5705560⎤⎥
⎥
⎥⎦
خطوة 2.3
Perform the row operation R3=R3-5R1R3=R3−5R1 to make the entry at 3,13,1 a 00.
خطوة 2.3.1
Perform the row operation R3=R3-5R1R3=R3−5R1 to make the entry at 3,13,1 a 00.
[1672700-157-5705-5⋅15-5(67)6-5(27)0-5⋅0]⎡⎢
⎢
⎢
⎢
⎢⎣1672700−157−5705−5⋅15−5(67)6−5(27)0−5⋅0⎤⎥
⎥
⎥
⎥
⎥⎦
خطوة 2.3.2
بسّط R3R3.
[1672700-157-5700573270]⎡⎢
⎢
⎢
⎢⎣1672700−157−5700573270⎤⎥
⎥
⎥
⎥⎦
[1672700-157-5700573270]⎡⎢
⎢
⎢
⎢⎣1672700−157−5700573270⎤⎥
⎥
⎥
⎥⎦
خطوة 2.4
Multiply each element of R2R2 by -715−715 to make the entry at 2,22,2 a 11.
خطوة 2.4.1
Multiply each element of R2R2 by -715−715 to make the entry at 2,22,2 a 11.
[167270-715⋅0-715(-157)-715(-57)-715⋅00573270]⎡⎢
⎢
⎢
⎢
⎢⎣167270−715⋅0−715(−157)−715(−57)−715⋅00573270⎤⎥
⎥
⎥
⎥
⎥⎦
خطوة 2.4.2
بسّط R2R2.
[167270011300573270]⎡⎢
⎢
⎢
⎢⎣167270011300573270⎤⎥
⎥
⎥
⎥⎦
[167270011300573270]⎡⎢
⎢
⎢
⎢⎣167270011300573270⎤⎥
⎥
⎥
⎥⎦
خطوة 2.5
Perform the row operation R3=R3-57R2R3=R3−57R2 to make the entry at 3,23,2 a 00.
خطوة 2.5.1
Perform the row operation R3=R3-57R2R3=R3−57R2 to make the entry at 3,23,2 a 00.
[167270011300-57⋅057-57⋅1327-57⋅130-57⋅0]⎡⎢
⎢
⎢
⎢⎣167270011300−57⋅057−57⋅1327−57⋅130−57⋅0⎤⎥
⎥
⎥
⎥⎦
خطوة 2.5.2
بسّط R3R3.
[16727001130001330]⎡⎢
⎢
⎢
⎢⎣16727001130001330⎤⎥
⎥
⎥
⎥⎦
[16727001130001330]⎡⎢
⎢
⎢
⎢⎣16727001130001330⎤⎥
⎥
⎥
⎥⎦
خطوة 2.6
Multiply each element of R3R3 by 313313 to make the entry at 3,33,3 a 11.
خطوة 2.6.1
Multiply each element of R3R3 by 313313 to make the entry at 3,33,3 a 11.
[16727001130313⋅0313⋅0313⋅133313⋅0]⎡⎢
⎢
⎢
⎢⎣16727001130313⋅0313⋅0313⋅133313⋅0⎤⎥
⎥
⎥
⎥⎦
خطوة 2.6.2
بسّط R3R3.
[167270011300010]⎡⎢
⎢
⎢⎣167270011300010⎤⎥
⎥
⎥⎦
[167270011300010]⎡⎢
⎢
⎢⎣167270011300010⎤⎥
⎥
⎥⎦
خطوة 2.7
Perform the row operation R2=R2-13R3R2=R2−13R3 to make the entry at 2,32,3 a 00.
خطوة 2.7.1
Perform the row operation R2=R2-13R3R2=R2−13R3 to make the entry at 2,32,3 a 00.
[1672700-13⋅01-13⋅013-13⋅10-13⋅00010]⎡⎢
⎢
⎢⎣1672700−13⋅01−13⋅013−13⋅10−13⋅00010⎤⎥
⎥
⎥⎦
خطوة 2.7.2
بسّط R2R2.
[16727001000010]⎡⎢
⎢⎣16727001000010⎤⎥
⎥⎦
[16727001000010]⎡⎢
⎢⎣16727001000010⎤⎥
⎥⎦
خطوة 2.8
Perform the row operation R1=R1-27R3R1=R1−27R3 to make the entry at 1,31,3 a 00.
خطوة 2.8.1
Perform the row operation R1=R1-27R3R1=R1−27R3 to make the entry at 1,31,3 a 00.
[1-27⋅067-27⋅027-27⋅10-27⋅001000010]⎡⎢
⎢⎣1−27⋅067−27⋅027−27⋅10−27⋅001000010⎤⎥
⎥⎦
خطوة 2.8.2
بسّط R1R1.
[1670001000010]⎡⎢
⎢⎣1670001000010⎤⎥
⎥⎦
[1670001000010]⎡⎢
⎢⎣1670001000010⎤⎥
⎥⎦
خطوة 2.9
Perform the row operation R1=R1-67R2R1=R1−67R2 to make the entry at 1,21,2 a 00.
خطوة 2.9.1
Perform the row operation R1=R1-67R2R1=R1−67R2 to make the entry at 1,21,2 a 00.
[1-67⋅067-67⋅10-67⋅00-67⋅001000010]⎡⎢
⎢⎣1−67⋅067−67⋅10−67⋅00−67⋅001000010⎤⎥
⎥⎦
خطوة 2.9.2
بسّط R1R1.
[100001000010]⎡⎢
⎢⎣100001000010⎤⎥
⎥⎦
[100001000010]⎡⎢
⎢⎣100001000010⎤⎥
⎥⎦
[100001000010]⎡⎢
⎢⎣100001000010⎤⎥
⎥⎦
خطوة 3
Use the result matrix to declare the final solution to the system of equations.
x=0x=0
y=0y=0
z=0z=0
خطوة 4
Write a solution vector by solving in terms of the free variables in each row.
[xyz]=[000]⎡⎢⎣xyz⎤⎥⎦=⎡⎢⎣000⎤⎥⎦
خطوة 5
Write as a solution set.
{[000]}⎧⎪⎨⎪⎩⎡⎢⎣000⎤⎥⎦⎫⎪⎬⎪⎭