Ví dụ
⎡⎢⎣330103020⎤⎥⎦
Bước 1
Bước 1.1
Choose the row or column with the most 0 elements. If there are no 0 elements choose any row or column. Multiply every element in row 3 by its cofactor and add.
Bước 1.1.1
Consider the corresponding sign chart.
∣∣
∣∣+−+−+−+−+∣∣
∣∣
Bước 1.1.2
The cofactor is the minor with the sign changed if the indices match a − position on the sign chart.
Bước 1.1.3
The minor for a31 is the determinant with row 3 and column 1 deleted.
∣∣∣3003∣∣∣
Bước 1.1.4
Multiply element a31 by its cofactor.
0∣∣∣3003∣∣∣
Bước 1.1.5
The minor for a32 is the determinant with row 3 and column 2 deleted.
∣∣∣3013∣∣∣
Bước 1.1.6
Multiply element a32 by its cofactor.
−2∣∣∣3013∣∣∣
Bước 1.1.7
The minor for a33 is the determinant with row 3 and column 3 deleted.
∣∣∣3310∣∣∣
Bước 1.1.8
Multiply element a33 by its cofactor.
0∣∣∣3310∣∣∣
Bước 1.1.9
Add the terms together.
0∣∣∣3003∣∣∣−2∣∣∣3013∣∣∣+0∣∣∣3310∣∣∣
0∣∣∣3003∣∣∣−2∣∣∣3013∣∣∣+0∣∣∣3310∣∣∣
Bước 1.2
Nhân 0 với ∣∣∣3003∣∣∣.
0−2∣∣∣3013∣∣∣+0∣∣∣3310∣∣∣
Bước 1.3
Nhân 0 với ∣∣∣3310∣∣∣.
0−2∣∣∣3013∣∣∣+0
Bước 1.4
Tính ∣∣∣3013∣∣∣.
Bước 1.4.1
Có thể tìm được định thức của một 2×2 ma trận bằng công thức ∣∣∣abcd∣∣∣=ad−cb.
0−2(3⋅3−1⋅0)+0
Bước 1.4.2
Rút gọn định thức.
Bước 1.4.2.1
Nhân 3 với 3.
0−2(9−1⋅0)+0
Bước 1.4.2.2
Trừ 0 khỏi 9.
0−2⋅9+0
0−2⋅9+0
0−2⋅9+0
Bước 1.5
Rút gọn định thức.
Bước 1.5.1
Nhân −2 với 9.
0−18+0
Bước 1.5.2
Trừ 18 khỏi 0.
−18+0
Bước 1.5.3
Cộng −18 và 0.
−18
−18
−18
Bước 2
Since the determinant is non-zero, the inverse exists.
Bước 3
Set up a 3×6 matrix where the left half is the original matrix and the right half is its identity matrix.
⎡⎢⎣330100103010020001⎤⎥⎦
Bước 4
Bước 4.1
Multiply each element of R1 by 13 to make the entry at 1,1 a 1.
Bước 4.1.1
Multiply each element of R1 by 13 to make the entry at 1,1 a 1.
⎡⎢
⎢⎣333303130303103010020001⎤⎥
⎥⎦
Bước 4.1.2
Rút gọn R1.
⎡⎢
⎢⎣1101300103010020001⎤⎥
⎥⎦
⎡⎢
⎢⎣1101300103010020001⎤⎥
⎥⎦
Bước 4.2
Perform the row operation R2=R2−R1 to make the entry at 2,1 a 0.
Bước 4.2.1
Perform the row operation R2=R2−R1 to make the entry at 2,1 a 0.
⎡⎢
⎢⎣11013001−10−13−00−131−00−0020001⎤⎥
⎥⎦
Bước 4.2.2
Rút gọn R2.
⎡⎢
⎢⎣11013000−13−1310020001⎤⎥
⎥⎦
⎡⎢
⎢⎣11013000−13−1310020001⎤⎥
⎥⎦
Bước 4.3
Multiply each element of R2 by −1 to make the entry at 2,2 a 1.
Bước 4.3.1
Multiply each element of R2 by −1 to make the entry at 2,2 a 1.
⎡⎢
⎢⎣1101300−0−−1−1⋅3−−13−1⋅1−0020001⎤⎥
⎥⎦
Bước 4.3.2
Rút gọn R2.
⎡⎢
⎢⎣110130001−313−10020001⎤⎥
⎥⎦
⎡⎢
⎢⎣110130001−313−10020001⎤⎥
⎥⎦
Bước 4.4
Perform the row operation R3=R3−2R2 to make the entry at 3,2 a 0.
Bước 4.4.1
Perform the row operation R3=R3−2R2 to make the entry at 3,2 a 0.
⎡⎢
⎢
⎢
⎢⎣110130001−313−100−2⋅02−2⋅10−2⋅−30−2(13)0−2⋅−11−2⋅0⎤⎥
⎥
⎥
⎥⎦
Bước 4.4.2
Rút gọn R3.
⎡⎢
⎢
⎢⎣110130001−313−10006−2321⎤⎥
⎥
⎥⎦
⎡⎢
⎢
⎢⎣110130001−313−10006−2321⎤⎥
⎥
⎥⎦
Bước 4.5
Multiply each element of R3 by 16 to make the entry at 3,3 a 1.
Bước 4.5.1
Multiply each element of R3 by 16 to make the entry at 3,3 a 1.
⎡⎢
⎢
⎢
⎢⎣110130001−313−10060666−2362616⎤⎥
⎥
⎥
⎥⎦
Bước 4.5.2
Rút gọn R3.
⎡⎢
⎢
⎢⎣110130001−313−10001−191316⎤⎥
⎥
⎥⎦
⎡⎢
⎢
⎢⎣110130001−313−10001−191316⎤⎥
⎥
⎥⎦
Bước 4.6
Perform the row operation R2=R2+3R3 to make the entry at 2,3 a 0.
Bước 4.6.1
Perform the row operation R2=R2+3R3 to make the entry at 2,3 a 0.
⎡⎢
⎢
⎢
⎢⎣11013000+3⋅01+3⋅0−3+3⋅113+3(−19)−1+3(13)0+3(16)001−191316⎤⎥
⎥
⎥
⎥⎦
Bước 4.6.2
Rút gọn R2.
⎡⎢
⎢
⎢⎣11013000100012001−191316⎤⎥
⎥
⎥⎦
⎡⎢
⎢
⎢⎣11013000100012001−191316⎤⎥
⎥
⎥⎦
Bước 4.7
Perform the row operation R1=R1−R2 to make the entry at 1,2 a 0.
Bước 4.7.1
Perform the row operation R1=R1−R2 to make the entry at 1,2 a 0.
⎡⎢
⎢
⎢⎣1−01−10−013−00−00−120100012001−191316⎤⎥
⎥
⎥⎦
Bước 4.7.2
Rút gọn R1.
⎡⎢
⎢
⎢⎣100130−120100012001−191316⎤⎥
⎥
⎥⎦
⎡⎢
⎢
⎢⎣100130−120100012001−191316⎤⎥
⎥
⎥⎦
⎡⎢
⎢
⎢⎣100130−120100012001−191316⎤⎥
⎥
⎥⎦
Bước 5
The right half of the reduced row echelon form is the inverse.
⎡⎢
⎢
⎢⎣130−120012−191316⎤⎥
⎥
⎥⎦
