الرياضيات المتناهية الأمثلة
[434112302]⎡⎢⎣434112302⎤⎥⎦
خطوة 1
خطوة 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 column 22 by its cofactor and add.
خطوة 1.1.1
Consider the corresponding sign chart.
|+-+-+-+-+|∣∣
∣∣+−+−+−+−+∣∣
∣∣
خطوة 1.1.2
The cofactor is the minor with the sign changed if the indices match a -− position on the sign chart.
خطوة 1.1.3
The minor for a12a12 is the determinant with row 11 and column 22 deleted.
|1232|∣∣∣1232∣∣∣
خطوة 1.1.4
Multiply element a12a12 by its cofactor.
-3|1232|−3∣∣∣1232∣∣∣
خطوة 1.1.5
The minor for a22a22 is the determinant with row 22 and column 22 deleted.
|4432|∣∣∣4432∣∣∣
خطوة 1.1.6
Multiply element a22a22 by its cofactor.
1|4432|1∣∣∣4432∣∣∣
خطوة 1.1.7
The minor for a32a32 is the determinant with row 33 and column 22 deleted.
|4412|∣∣∣4412∣∣∣
خطوة 1.1.8
Multiply element a32a32 by its cofactor.
0|4412|0∣∣∣4412∣∣∣
خطوة 1.1.9
Add the terms together.
-3|1232|+1|4432|+0|4412|−3∣∣∣1232∣∣∣+1∣∣∣4432∣∣∣+0∣∣∣4412∣∣∣
-3|1232|+1|4432|+0|4412|−3∣∣∣1232∣∣∣+1∣∣∣4432∣∣∣+0∣∣∣4412∣∣∣
خطوة 1.2
اضرب 00 في |4412|∣∣∣4412∣∣∣.
-3|1232|+1|4432|+0−3∣∣∣1232∣∣∣+1∣∣∣4432∣∣∣+0
خطوة 1.3
احسِب قيمة |1232|∣∣∣1232∣∣∣.
خطوة 1.3.1
يمكن إيجاد محدد المصفوفة 2×22×2 باستخدام القاعدة |abcd|=ad-cb∣∣∣abcd∣∣∣=ad−cb.
-3(1⋅2-3⋅2)+1|4432|+0−3(1⋅2−3⋅2)+1∣∣∣4432∣∣∣+0
خطوة 1.3.2
بسّط المحدد.
خطوة 1.3.2.1
بسّط كل حد.
خطوة 1.3.2.1.1
اضرب 22 في 11.
-3(2-3⋅2)+1|4432|+0−3(2−3⋅2)+1∣∣∣4432∣∣∣+0
خطوة 1.3.2.1.2
اضرب -3−3 في 22.
-3(2-6)+1|4432|+0−3(2−6)+1∣∣∣4432∣∣∣+0
-3(2-6)+1|4432|+0−3(2−6)+1∣∣∣4432∣∣∣+0
خطوة 1.3.2.2
اطرح 66 من 22.
-3⋅-4+1|4432|+0−3⋅−4+1∣∣∣4432∣∣∣+0
-3⋅-4+1|4432|+0−3⋅−4+1∣∣∣4432∣∣∣+0
-3⋅-4+1|4432|+0−3⋅−4+1∣∣∣4432∣∣∣+0
خطوة 1.4
احسِب قيمة |4432|∣∣∣4432∣∣∣.
خطوة 1.4.1
يمكن إيجاد محدد المصفوفة 2×22×2 باستخدام القاعدة |abcd|=ad-cb∣∣∣abcd∣∣∣=ad−cb.
-3⋅-4+1(4⋅2-3⋅4)+0−3⋅−4+1(4⋅2−3⋅4)+0
خطوة 1.4.2
بسّط المحدد.
خطوة 1.4.2.1
بسّط كل حد.
خطوة 1.4.2.1.1
اضرب 44 في 22.
-3⋅-4+1(8-3⋅4)+0−3⋅−4+1(8−3⋅4)+0
خطوة 1.4.2.1.2
اضرب -3−3 في 44.
-3⋅-4+1(8-12)+0−3⋅−4+1(8−12)+0
-3⋅-4+1(8-12)+0−3⋅−4+1(8−12)+0
خطوة 1.4.2.2
اطرح 1212 من 88.
-3⋅-4+1⋅-4+0−3⋅−4+1⋅−4+0
-3⋅-4+1⋅-4+0−3⋅−4+1⋅−4+0
-3⋅-4+1⋅-4+0−3⋅−4+1⋅−4+0
خطوة 1.5
بسّط المحدد.
خطوة 1.5.1
بسّط كل حد.
خطوة 1.5.1.1
اضرب -3−3 في -4−4.
12+1⋅-4+012+1⋅−4+0
خطوة 1.5.1.2
اضرب -4−4 في 11.
12-4+012−4+0
12-4+012−4+0
خطوة 1.5.2
اطرح 44 من 1212.
8+08+0
خطوة 1.5.3
أضف 88 و00.
88
88
88
خطوة 2
Since the determinant is non-zero, the inverse exists.
خطوة 3
Set up a 3×63×6 matrix where the left half is the original matrix and the right half is its identity matrix.
[434100112010302001]⎡⎢⎣434100112010302001⎤⎥⎦
خطوة 4
خطوة 4.1
Multiply each element of R1R1 by 1414 to make the entry at 1,11,1 a 11.
خطوة 4.1.1
Multiply each element of R1R1 by 1414 to make the entry at 1,11,1 a 11.
[443444140404112010302001]⎡⎢
⎢⎣443444140404112010302001⎤⎥
⎥⎦
خطوة 4.1.2
بسّط R1R1.
[13411400112010302001]⎡⎢
⎢⎣13411400112010302001⎤⎥
⎥⎦
[13411400112010302001]⎡⎢
⎢⎣13411400112010302001⎤⎥
⎥⎦
خطوة 4.2
Perform the row operation R2=R2-R1R2=R2−R1 to make the entry at 2,12,1 a 00.
خطوة 4.2.1
Perform the row operation R2=R2-R1R2=R2−R1 to make the entry at 2,12,1 a 00.
[134114001-11-342-10-141-00-0302001]⎡⎢
⎢⎣134114001−11−342−10−141−00−0302001⎤⎥
⎥⎦
خطوة 4.2.2
بسّط R2R2.
[134114000141-1410302001]⎡⎢
⎢⎣134114000141−1410302001⎤⎥
⎥⎦
[134114000141-1410302001]⎡⎢
⎢⎣134114000141−1410302001⎤⎥
⎥⎦
خطوة 4.3
Perform the row operation R3=R3-3R1R3=R3−3R1 to make the entry at 3,13,1 a 00.
خطوة 4.3.1
Perform the row operation R3=R3-3R1R3=R3−3R1 to make the entry at 3,13,1 a 00.
[134114000141-14103-3⋅10-3(34)2-3⋅10-3(14)0-3⋅01-3⋅0]⎡⎢
⎢
⎢
⎢⎣134114000141−14103−3⋅10−3(34)2−3⋅10−3(14)0−3⋅01−3⋅0⎤⎥
⎥
⎥
⎥⎦
خطوة 4.3.2
بسّط R3R3.
[134114000141-14100-94-1-3401]⎡⎢
⎢
⎢⎣134114000141−14100−94−1−3401⎤⎥
⎥
⎥⎦
[134114000141-14100-94-1-3401]⎡⎢
⎢
⎢⎣134114000141−14100−94−1−3401⎤⎥
⎥
⎥⎦
خطوة 4.4
Multiply each element of R2R2 by 44 to make the entry at 2,22,2 a 11.
خطوة 4.4.1
Multiply each element of R2R2 by 44 to make the entry at 2,22,2 a 11.
[134114004⋅04(14)4⋅14(-14)4⋅14⋅00-94-1-3401]⎡⎢
⎢
⎢⎣134114004⋅04(14)4⋅14(−14)4⋅14⋅00−94−1−3401⎤⎥
⎥
⎥⎦
خطوة 4.4.2
بسّط R2R2.
[13411400014-1400-94-1-3401]⎡⎢
⎢⎣13411400014−1400−94−1−3401⎤⎥
⎥⎦
[13411400014-1400-94-1-3401]⎡⎢
⎢⎣13411400014−1400−94−1−3401⎤⎥
⎥⎦
خطوة 4.5
Perform the row operation R3=R3+94R2R3=R3+94R2 to make the entry at 3,23,2 a 00.
خطوة 4.5.1
Perform the row operation R3=R3+94R2R3=R3+94R2 to make the entry at 3,23,2 a 00.
[13411400014-1400+94⋅0-94+94⋅1-1+94⋅4-34+94⋅-10+94⋅41+94⋅0]⎡⎢
⎢⎣13411400014−1400+94⋅0−94+94⋅1−1+94⋅4−34+94⋅−10+94⋅41+94⋅0⎤⎥
⎥⎦
خطوة 4.5.2
بسّط R3R3.
[13411400014-140008-391]⎡⎢
⎢⎣13411400014−140008−391⎤⎥
⎥⎦
[13411400014-140008-391]⎡⎢
⎢⎣13411400014−140008−391⎤⎥
⎥⎦
خطوة 4.6
Multiply each element of R3R3 by 1818 to make the entry at 3,33,3 a 11.
خطوة 4.6.1
Multiply each element of R3R3 by 1818 to make the entry at 3,33,3 a 11.
[13411400014-140080888-389818]⎡⎢
⎢⎣13411400014−140080888−389818⎤⎥
⎥⎦
خطوة 4.6.2
بسّط R3R3.
[13411400014-140001-389818]⎡⎢
⎢⎣13411400014−140001−389818⎤⎥
⎥⎦
[13411400014-140001-389818]⎡⎢
⎢⎣13411400014−140001−389818⎤⎥
⎥⎦
خطوة 4.7
Perform the row operation R2=R2-4R3R2=R2−4R3 to make the entry at 2,32,3 a 00.
خطوة 4.7.1
Perform the row operation R2=R2-4R3R2=R2−4R3 to make the entry at 2,32,3 a 00.
[134114000-4⋅01-4⋅04-4⋅1-1-4(-38)4-4(98)0-4(18)001-389818]⎡⎢
⎢
⎢
⎢⎣134114000−4⋅01−4⋅04−4⋅1−1−4(−38)4−4(98)0−4(18)001−389818⎤⎥
⎥
⎥
⎥⎦
خطوة 4.7.2
بسّط R2R2.
[1341140001012-12-12001-389818]⎡⎢
⎢
⎢⎣1341140001012−12−12001−389818⎤⎥
⎥
⎥⎦
[1341140001012-12-12001-389818]⎡⎢
⎢
⎢⎣1341140001012−12−12001−389818⎤⎥
⎥
⎥⎦
خطوة 4.8
Perform the row operation R1=R1-R3R1=R1−R3 to make the entry at 1,31,3 a 00.
خطوة 4.8.1
Perform the row operation R1=R1-R3R1=R1−R3 to make the entry at 1,31,3 a 00.
[1-034-01-114+380-980-1801012-12-12001-389818]⎡⎢
⎢
⎢⎣1−034−01−114+380−980−1801012−12−12001−389818⎤⎥
⎥
⎥⎦
خطوة 4.8.2
بسّط R1R1.
[134058-98-1801012-12-12001-389818]⎡⎢
⎢
⎢⎣134058−98−1801012−12−12001−389818⎤⎥
⎥
⎥⎦
[134058-98-1801012-12-12001-389818]⎡⎢
⎢
⎢⎣134058−98−1801012−12−12001−389818⎤⎥
⎥
⎥⎦
خطوة 4.9
Perform the row operation R1=R1-34R2R1=R1−34R2 to make the entry at 1,21,2 a 00.
خطوة 4.9.1
Perform the row operation R1=R1-34R2R1=R1−34R2 to make the entry at 1,21,2 a 00.
[1-34⋅034-34⋅10-34⋅058-34⋅12-98-34(-12)-18-34(-12)01012-12-12001-389818]⎡⎢
⎢
⎢⎣1−34⋅034−34⋅10−34⋅058−34⋅12−98−34(−12)−18−34(−12)01012−12−12001−389818⎤⎥
⎥
⎥⎦
خطوة 4.9.2
بسّط R1R1.
[10014-341401012-12-12001-389818]⎡⎢
⎢
⎢⎣10014−341401012−12−12001−389818⎤⎥
⎥
⎥⎦
[10014-341401012-12-12001-389818]
[10014-341401012-12-12001-389818]
خطوة 5
The right half of the reduced row echelon form is the inverse.
[14-341412-12-12-389818]
