Aljabar Linear Contoh
[413144441]⎡⎢⎣413144441⎤⎥⎦
Langkah 1
Langkah 1.1
Choose the row or column with the most 00 elements. If there are no 00 elements choose any row or column. Multiply every element in row 11 by its cofactor and add.
Langkah 1.1.1
Consider the corresponding sign chart.
|+-+-+-+-+|∣∣
∣∣+−+−+−+−+∣∣
∣∣
Langkah 1.1.2
The cofactor is the minor with the sign changed if the indices match a -− position on the sign chart.
Langkah 1.1.3
The minor for a11a11 is the determinant with row 11 and column 11 deleted.
|4441|∣∣∣4441∣∣∣
Langkah 1.1.4
Multiply element a11a11 by its cofactor.
4|4441|4∣∣∣4441∣∣∣
Langkah 1.1.5
The minor for a12a12 is the determinant with row 11 and column 22 deleted.
|1441|∣∣∣1441∣∣∣
Langkah 1.1.6
Multiply element a12a12 by its cofactor.
-1|1441|−1∣∣∣1441∣∣∣
Langkah 1.1.7
The minor for a13a13 is the determinant with row 11 and column 33 deleted.
|1444|∣∣∣1444∣∣∣
Langkah 1.1.8
Multiply element a13a13 by its cofactor.
3|1444|3∣∣∣1444∣∣∣
Langkah 1.1.9
Add the terms together.
4|4441|-1|1441|+3|1444|4∣∣∣4441∣∣∣−1∣∣∣1441∣∣∣+3∣∣∣1444∣∣∣
4|4441|-1|1441|+3|1444|4∣∣∣4441∣∣∣−1∣∣∣1441∣∣∣+3∣∣∣1444∣∣∣
Langkah 1.2
Evaluasi |4441|∣∣∣4441∣∣∣.
Langkah 1.2.1
Determinan dari matriks 2×22×2 dapat dicari menggunakan rumus |abcd|=ad-cb∣∣∣abcd∣∣∣=ad−cb.
4(4⋅1-4⋅4)-1|1441|+3|1444|4(4⋅1−4⋅4)−1∣∣∣1441∣∣∣+3∣∣∣1444∣∣∣
Langkah 1.2.2
Sederhanakan determinannya.
Langkah 1.2.2.1
Sederhanakan setiap suku.
Langkah 1.2.2.1.1
Kalikan 44 dengan 11.
4(4-4⋅4)-1|1441|+3|1444|4(4−4⋅4)−1∣∣∣1441∣∣∣+3∣∣∣1444∣∣∣
Langkah 1.2.2.1.2
Kalikan -4−4 dengan 44.
4(4-16)-1|1441|+3|1444|4(4−16)−1∣∣∣1441∣∣∣+3∣∣∣1444∣∣∣
4(4-16)-1|1441|+3|1444|4(4−16)−1∣∣∣1441∣∣∣+3∣∣∣1444∣∣∣
Langkah 1.2.2.2
Kurangi 1616 dengan 44.
4⋅-12-1|1441|+3|1444|4⋅−12−1∣∣∣1441∣∣∣+3∣∣∣1444∣∣∣
4⋅-12-1|1441|+3|1444|4⋅−12−1∣∣∣1441∣∣∣+3∣∣∣1444∣∣∣
4⋅-12-1|1441|+3|1444|4⋅−12−1∣∣∣1441∣∣∣+3∣∣∣1444∣∣∣
Langkah 1.3
Evaluasi |1441|∣∣∣1441∣∣∣.
Langkah 1.3.1
Determinan dari matriks 2×22×2 dapat dicari menggunakan rumus |abcd|=ad-cb∣∣∣abcd∣∣∣=ad−cb.
4⋅-12-1(1⋅1-4⋅4)+3|1444|4⋅−12−1(1⋅1−4⋅4)+3∣∣∣1444∣∣∣
Langkah 1.3.2
Sederhanakan determinannya.
Langkah 1.3.2.1
Sederhanakan setiap suku.
Langkah 1.3.2.1.1
Kalikan 11 dengan 11.
4⋅-12-1(1-4⋅4)+3|1444|4⋅−12−1(1−4⋅4)+3∣∣∣1444∣∣∣
Langkah 1.3.2.1.2
Kalikan -4−4 dengan 44.
4⋅-12-1(1-16)+3|1444|4⋅−12−1(1−16)+3∣∣∣1444∣∣∣
4⋅-12-1(1-16)+3|1444|4⋅−12−1(1−16)+3∣∣∣1444∣∣∣
Langkah 1.3.2.2
Kurangi 1616 dengan 11.
4⋅-12-1⋅-15+3|1444|4⋅−12−1⋅−15+3∣∣∣1444∣∣∣
4⋅-12-1⋅-15+3|1444|4⋅−12−1⋅−15+3∣∣∣1444∣∣∣
4⋅-12-1⋅-15+3|1444|4⋅−12−1⋅−15+3∣∣∣1444∣∣∣
Langkah 1.4
Evaluasi |1444|∣∣∣1444∣∣∣.
Langkah 1.4.1
Determinan dari matriks 2×22×2 dapat dicari menggunakan rumus |abcd|=ad-cb∣∣∣abcd∣∣∣=ad−cb.
4⋅-12-1⋅-15+3(1⋅4-4⋅4)4⋅−12−1⋅−15+3(1⋅4−4⋅4)
Langkah 1.4.2
Sederhanakan determinannya.
Langkah 1.4.2.1
Sederhanakan setiap suku.
Langkah 1.4.2.1.1
Kalikan 44 dengan 11.
4⋅-12-1⋅-15+3(4-4⋅4)4⋅−12−1⋅−15+3(4−4⋅4)
Langkah 1.4.2.1.2
Kalikan -4−4 dengan 44.
4⋅-12-1⋅-15+3(4-16)4⋅−12−1⋅−15+3(4−16)
4⋅-12-1⋅-15+3(4-16)4⋅−12−1⋅−15+3(4−16)
Langkah 1.4.2.2
Kurangi 1616 dengan 44.
4⋅-12-1⋅-15+3⋅-124⋅−12−1⋅−15+3⋅−12
4⋅-12-1⋅-15+3⋅-124⋅−12−1⋅−15+3⋅−12
4⋅-12-1⋅-15+3⋅-124⋅−12−1⋅−15+3⋅−12
Langkah 1.5
Sederhanakan determinannya.
Langkah 1.5.1
Sederhanakan setiap suku.
Langkah 1.5.1.1
Kalikan 44 dengan -12−12.
-48-1⋅-15+3⋅-12−48−1⋅−15+3⋅−12
Langkah 1.5.1.2
Kalikan -1−1 dengan -15−15.
-48+15+3⋅-12−48+15+3⋅−12
Langkah 1.5.1.3
Kalikan 33 dengan -12−12.
-48+15-36−48+15−36
-48+15-36−48+15−36
Langkah 1.5.2
Tambahkan -48−48 dan 1515.
-33-36−33−36
Langkah 1.5.3
Kurangi 3636 dengan -33−33.
-69−69
-69−69
-69−69
Langkah 2
Since the determinant is non-zero, the inverse exists.
Langkah 3
Set up a 3×63×6 matrix where the left half is the original matrix and the right half is its identity matrix.
[413100144010441001]⎡⎢⎣413100144010441001⎤⎥⎦
Langkah 4
Langkah 4.1
Multiply each element of R1R1 by 1414 to make the entry at 1,11,1 a 11.
Langkah 4.1.1
Multiply each element of R1R1 by 1414 to make the entry at 1,11,1 a 11.
[441434140404144010441001]⎡⎢
⎢⎣441434140404144010441001⎤⎥
⎥⎦
Langkah 4.1.2
Sederhanakan R1R1.
[114341400144010441001]⎡⎢
⎢⎣114341400144010441001⎤⎥
⎥⎦
[114341400144010441001]⎡⎢
⎢⎣114341400144010441001⎤⎥
⎥⎦
Langkah 4.2
Perform the row operation R2=R2-R1R2=R2−R1 to make the entry at 2,12,1 a 00.
Langkah 4.2.1
Perform the row operation R2=R2-R1R2=R2−R1 to make the entry at 2,12,1 a 00.
[1143414001-14-144-340-141-00-0441001]⎡⎢
⎢⎣1143414001−14−144−340−141−00−0441001⎤⎥
⎥⎦
Langkah 4.2.2
Sederhanakan R2R2.
[1143414000154134-1410441001]⎡⎢
⎢⎣1143414000154134−1410441001⎤⎥
⎥⎦
[1143414000154134-1410441001]⎡⎢
⎢⎣1143414000154134−1410441001⎤⎥
⎥⎦
Langkah 4.3
Perform the row operation R3=R3-4R1R3=R3−4R1 to make the entry at 3,13,1 a 00.
Langkah 4.3.1
Perform the row operation R3=R3-4R1R3=R3−4R1 to make the entry at 3,13,1 a 00.
[1143414000154134-14104-4⋅14-4(14)1-4(34)0-4(14)0-4⋅01-4⋅0]⎡⎢
⎢
⎢
⎢⎣1143414000154134−14104−4⋅14−4(14)1−4(34)0−4(14)0−4⋅01−4⋅0⎤⎥
⎥
⎥
⎥⎦
Langkah 4.3.2
Sederhanakan R3R3.
[1143414000154134-141003-2-101]⎡⎢
⎢⎣1143414000154134−141003−2−101⎤⎥
⎥⎦
[1143414000154134-141003-2-101]⎡⎢
⎢⎣1143414000154134−141003−2−101⎤⎥
⎥⎦
Langkah 4.4
Multiply each element of R2R2 by 415415 to make the entry at 2,22,2 a 11.
Langkah 4.4.1
Multiply each element of R2R2 by 415415 to make the entry at 2,22,2 a 11.
[114341400415⋅0415⋅154415⋅134415(-14)415⋅1415⋅003-2-101]⎡⎢
⎢⎣114341400415⋅0415⋅154415⋅134415(−14)415⋅1415⋅003−2−101⎤⎥
⎥⎦
Langkah 4.4.2
Sederhanakan R2R2.
[114341400011315-115415003-2-101]⎡⎢
⎢⎣114341400011315−115415003−2−101⎤⎥
⎥⎦
[114341400011315-115415003-2-101]⎡⎢
⎢⎣114341400011315−115415003−2−101⎤⎥
⎥⎦
Langkah 4.5
Perform the row operation R3=R3-3R2R3=R3−3R2 to make the entry at 3,23,2 a 00.
Langkah 4.5.1
Perform the row operation R3=R3-3R2R3=R3−3R2 to make the entry at 3,23,2 a 00.
[114341400011315-11541500-3⋅03-3⋅1-2-3(1315)-1-3(-115)0-3(415)1-3⋅0]⎡⎢
⎢
⎢
⎢⎣114341400011315−11541500−3⋅03−3⋅1−2−3(1315)−1−3(−115)0−3(415)1−3⋅0⎤⎥
⎥
⎥
⎥⎦
Langkah 4.5.2
Sederhanakan R3R3.
[114341400011315-115415000-235-45-451]⎡⎢
⎢
⎢⎣114341400011315−115415000−235−45−451⎤⎥
⎥
⎥⎦
[114341400011315-115415000-235-45-451]⎡⎢
⎢
⎢⎣114341400011315−115415000−235−45−451⎤⎥
⎥
⎥⎦
Langkah 4.6
Multiply each element of R3R3 by -523−523 to make the entry at 3,33,3 a 11.
Langkah 4.6.1
Multiply each element of R3R3 by -523−523 to make the entry at 3,33,3 a 11.
[114341400011315-1154150-523⋅0-523⋅0-523(-235)-523(-45)-523(-45)-523⋅1]⎡⎢
⎢
⎢
⎢⎣114341400011315−1154150−523⋅0−523⋅0−523(−235)−523(−45)−523(−45)−523⋅1⎤⎥
⎥
⎥
⎥⎦
Langkah 4.6.2
Sederhanakan R3R3.
[114341400011315-1154150001423423-523]⎡⎢
⎢
⎢⎣114341400011315−1154150001423423−523⎤⎥
⎥
⎥⎦
[114341400011315-1154150001423423-523]⎡⎢
⎢
⎢⎣114341400011315−1154150001423423−523⎤⎥
⎥
⎥⎦
Langkah 4.7
Perform the row operation R2=R2-1315R3R2=R2−1315R3 to make the entry at 2,32,3 a 00.
Langkah 4.7.1
Perform the row operation R2=R2-1315R3R2=R2−1315R3 to make the entry at 2,32,3 a 00.
[1143414000-1315⋅01-1315⋅01315-1315⋅1-115-1315⋅423415-1315⋅4230-1315(-523)001423423-523]⎡⎢
⎢
⎢
⎢⎣1143414000−1315⋅01−1315⋅01315−1315⋅1−115−1315⋅423415−1315⋅4230−1315(−523)001423423−523⎤⎥
⎥
⎥
⎥⎦
Langkah 4.7.2
Sederhanakan R2R2.
[114341400010-5238691369001423423-523]⎡⎢
⎢
⎢⎣114341400010−5238691369001423423−523⎤⎥
⎥
⎥⎦
[114341400010-5238691369001423423-523]⎡⎢
⎢
⎢⎣114341400010−5238691369001423423−523⎤⎥
⎥
⎥⎦
Langkah 4.8
Perform the row operation R1=R1-34R3R1=R1−34R3 to make the entry at 1,31,3 a 00.
Langkah 4.8.1
Perform the row operation R1=R1-34R3R1=R1−34R3 to make the entry at 1,31,3 a 00.
[1-34⋅014-34⋅034-34⋅114-34⋅4230-34⋅4230-34(-523)010-5238691369001423423-523]⎡⎢
⎢
⎢
⎢⎣1−34⋅014−34⋅034−34⋅114−34⋅4230−34⋅4230−34(−523)010−5238691369001423423−523⎤⎥
⎥
⎥
⎥⎦
Langkah 4.8.2
Sederhanakan R1R1.
[11401192-3231592010-5238691369001423423-523]⎡⎢
⎢
⎢⎣11401192−3231592010−5238691369001423423−523⎤⎥
⎥
⎥⎦
[11401192-3231592010-5238691369001423423-523]⎡⎢
⎢
⎢⎣11401192−3231592010−5238691369001423423−523⎤⎥
⎥
⎥⎦
Langkah 4.9
Perform the row operation R1=R1-14R2R1=R1−14R2 to make the entry at 1,21,2 a 00.
Langkah 4.9.1
Perform the row operation R1=R1-14R2R1=R1−14R2 to make the entry at 1,21,2 a 00.
[1-14⋅014-14⋅10-14⋅01192-14(-523)-323-14⋅8691592-14⋅1369010-5238691369001423423-523]⎡⎢
⎢
⎢
⎢⎣1−14⋅014−14⋅10−14⋅01192−14(−523)−323−14⋅8691592−14⋅1369010−5238691369001423423−523⎤⎥
⎥
⎥
⎥⎦
Langkah 4.9.2
Sederhanakan R1R1.
[100423-1169869010-5238691369001423423-523]⎡⎢
⎢
⎢⎣100423−1169869010−5238691369001423423−523⎤⎥
⎥
⎥⎦
[100423-1169869010-5238691369001423423-523]
[100423-1169869010-5238691369001423423-523]
Langkah 5
The right half of the reduced row echelon form is the inverse.
[423-1169869-5238691369423423-523]
