Lineare Algebra Beispiele
[24681012120]⎡⎢⎣24681012120⎤⎥⎦
Schritt 1
Schritt 1.1
Multiply each element of R1R1 by 1212 to make the entry at 1,11,1 a 11.
Schritt 1.1.1
Multiply each element of R1R1 by 1212 to make the entry at 1,11,1 a 11.
[22426281012120]⎡⎢
⎢⎣22426281012120⎤⎥
⎥⎦
Schritt 1.1.2
Vereinfache R1R1.
[12381012120]⎡⎢⎣12381012120⎤⎥⎦
[12381012120]⎡⎢⎣12381012120⎤⎥⎦
Schritt 1.2
Perform the row operation R2=R2-8R1R2=R2−8R1 to make the entry at 2,12,1 a 00.
Schritt 1.2.1
Perform the row operation R2=R2-8R1R2=R2−8R1 to make the entry at 2,12,1 a 00.
[1238-8⋅110-8⋅212-8⋅3120]⎡⎢⎣1238−8⋅110−8⋅212−8⋅3120⎤⎥⎦
Schritt 1.2.2
Vereinfache R2R2.
[1230-6-12120]⎡⎢⎣1230−6−12120⎤⎥⎦
[1230-6-12120]⎡⎢⎣1230−6−12120⎤⎥⎦
Schritt 1.3
Perform the row operation R3=R3-R1R3=R3−R1 to make the entry at 3,13,1 a 00.
Schritt 1.3.1
Perform the row operation R3=R3-R1R3=R3−R1 to make the entry at 3,13,1 a 00.
[1230-6-121-12-20-3]⎡⎢⎣1230−6−121−12−20−3⎤⎥⎦
Schritt 1.3.2
Vereinfache R3R3.
[1230-6-1200-3]⎡⎢⎣1230−6−1200−3⎤⎥⎦
[1230-6-1200-3]⎡⎢⎣1230−6−1200−3⎤⎥⎦
Schritt 1.4
Multiply each element of R2R2 by -16−16 to make the entry at 2,22,2 a 11.
Schritt 1.4.1
Multiply each element of R2R2 by -16−16 to make the entry at 2,22,2 a 11.
[123-16⋅0-16⋅-6-16⋅-1200-3]⎡⎢
⎢⎣123−16⋅0−16⋅−6−16⋅−1200−3⎤⎥
⎥⎦
Schritt 1.4.2
Vereinfache R2R2.
[12301200-3]⎡⎢⎣12301200−3⎤⎥⎦
[12301200-3]⎡⎢⎣12301200−3⎤⎥⎦
Schritt 1.5
Multiply each element of R3R3 by -13−13 to make the entry at 3,33,3 a 11.
Schritt 1.5.1
Multiply each element of R3R3 by -13−13 to make the entry at 3,33,3 a 11.
[123012-13⋅0-13⋅0-13⋅-3]⎡⎢
⎢⎣123012−13⋅0−13⋅0−13⋅−3⎤⎥
⎥⎦
Schritt 1.5.2
Vereinfache R3R3.
[123012001]⎡⎢⎣123012001⎤⎥⎦
[123012001]⎡⎢⎣123012001⎤⎥⎦
Schritt 1.6
Perform the row operation R2=R2-2R3R2=R2−2R3 to make the entry at 2,32,3 a 00.
Schritt 1.6.1
Perform the row operation R2=R2-2R3R2=R2−2R3 to make the entry at 2,32,3 a 00.
[1230-2⋅01-2⋅02-2⋅1001]⎡⎢⎣1230−2⋅01−2⋅02−2⋅1001⎤⎥⎦
Schritt 1.6.2
Vereinfache R2R2.
[123010001]⎡⎢⎣123010001⎤⎥⎦
[123010001]⎡⎢⎣123010001⎤⎥⎦
Schritt 1.7
Perform the row operation R1=R1-3R3R1=R1−3R3 to make the entry at 1,31,3 a 00.
Schritt 1.7.1
Perform the row operation R1=R1-3R3R1=R1−3R3 to make the entry at 1,31,3 a 00.
[1-3⋅02-3⋅03-3⋅1010001]⎡⎢⎣1−3⋅02−3⋅03−3⋅1010001⎤⎥⎦
Schritt 1.7.2
Vereinfache R1R1.
[120010001]⎡⎢⎣120010001⎤⎥⎦
[120010001]⎡⎢⎣120010001⎤⎥⎦
Schritt 1.8
Perform the row operation R1=R1-2R2R1=R1−2R2 to make the entry at 1,21,2 a 00.
Schritt 1.8.1
Perform the row operation R1=R1-2R2R1=R1−2R2 to make the entry at 1,21,2 a 00.
[1-2⋅02-2⋅10-2⋅0010001]⎡⎢⎣1−2⋅02−2⋅10−2⋅0010001⎤⎥⎦
Schritt 1.8.2
Vereinfache R1R1.
[100010001]⎡⎢⎣100010001⎤⎥⎦
[100010001]⎡⎢⎣100010001⎤⎥⎦
[100010001]⎡⎢⎣100010001⎤⎥⎦
Schritt 2
The pivot positions are the locations with the leading 11 in each row. The pivot columns are the columns that have a pivot position.
Pivot Positions: a11,a22,a11,a22, and a33a33
Pivot Columns: 1,2,1,2, and 33