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3131313131351515151515≈83137
How to find the inverse by Gaussian elimination with back-substitution
Augment the matrix with the identity to form [A|I]. Apply forward elimination to reduce A to upper-triangular form, propagating each row operation through the identity side. Then use back-substitution to clear the upper triangle. When A reaches the identity, the right side holds the inverse.
Gaussian elimination inverse — worked example (4×4)
Write the initial matrix
A
:
A
=
2
1
0
3
1
4
2
0
3
0
5
1
0
2
1
4
To find the inverse matrix of matrix
A
, we can add to the right of it the identity matrix of the same size;
After that, using the
Gaussian elimination
method, we transform the matrix so that the left part becomes a identity matrix, then in the right part we get the inverse matrix of the matrix
A
;
Write the expanded matrix (adding the identity matrix to the right of matrix
A
):
2
1
0
3
1
4
2
0
3
0
5
1
0
2
1
4
1
0
0
0
0
1
0
0
0
0
1
0
0
0
0
1
Gaussian Run Forwards
3
Iteration 1Divide
1
th row by
2
;
1
1
0
3
1
2
4
2
0
1
1
2
0
5
1
0
2
1
4
1
2
0
0
0
0
1
0
0
0
0
1
0
0
0
0
1
a
0
1,1
=
2
2
=
1
;
a
0
1,2
=
1
2
=
1
2
;
a
0
1,3
=
3
2
=
1
1
2
;
a
0
1,4
=
0
2
=
0
;
a
0
1,5
=
1
2
=
1
2
;
a
0
1,6
=
0
2
=
0
;
a
0
1,7
=
0
2
=
0
;
a
0
1,8
=
0
2
=
0
;
Hide description
From
2
th row we subtract
1
th row;
From
4
th row we subtract
1
th row, multiplied by
3
;
1
0
0
0
1
2
3
1
2
2
-1
1
2
1
1
2
-1
1
2
5
-3
1
2
0
2
1
4
1
2
-
1
2
0
-1
1
2
0
1
0
0
0
0
1
0
0
0
0
1
a
0
2,1
=
1
-
1
=
0
;
a
0
2,2
=
4
-
1
2
=
3
1
2
;
a
0
2,3
=
0
-
1
1
2
=
-1
1
2
;
a
0
2,4
=
2
-
0
=
2
;
a
0
2,5
=
0
-
1
2
=
-
1
2
;
a
0
2,6
=
1
-
0
=
1
;
a
0
2,7
=
0
-
0
=
0
;
a
0
2,8
=
0
-
0
=
0
;
a
0
4,1
=
3
- (
3
*
1
) =
0
;
a
0
4,2
=
0
- (
3
*
1
2
) =
-1
1
2
;
a
0
4,3
=
1
- (
3
*
1
1
2
) =
-3
1
2
;
a
0
4,4
=
4
- (
3
*
0
) =
4
;
a
0
4,5
=
0
- (
3
*
1
2
) =
-1
1
2
;
a
0
4,6
=
0
- (
3
*
0
) =
0
;
a
0
4,7
=
0
- (
3
*
0
) =
0
;
a
0
4,8
=
1
- (
3
*
0
) =
1
;
Hide description
4
Iteration 2Divide
2
th row by
3
1
2
;
1
0
0
0
1
2
1
2
-1
1
2
1
1
2
-
3
7
5
-3
1
2
0
4
7
1
4
1
2
-
1
7
0
-1
1
2
0
2
7
0
0
0
0
1
0
0
0
0
1
a
0
2,2
=
3
1
2
3
1
2
=
1
;
a
0
2,3
=
-1
1
2
3
1
2
=
-
3
7
;
a
0
2,4
=
2
3
1
2
=
4
7
;
a
0
2,5
=
-
1
2
3
1
2
=
-
1
7
;
a
0
2,6
=
1
3
1
2
=
2
7
;
a
0
2,7
=
0
3
1
2
=
0
;
a
0
2,8
=
0
3
1
2
=
0
;
Hide description
From
3
th row we subtract
2
th row, multiplied by
2
;
From
4
th row we subtract
2
th row, multiplied by
-1
1
2
;
1
0
0
0
1
2
1
0
0
1
1
2
-
3
7
5
6
7
-4
1
7
0
4
7
-
1
7
4
6
7
1
2
-
1
7
2
7
-1
5
7
0
2
7
-
4
7
3
7
0
0
1
0
0
0
0
1
a
0
3,2
=
2
- (
2
*
1
) =
0
;
a
0
3,3
=
5
- (
2
*
-
3
7
) =
5
6
7
;
a
0
3,4
=
1
- (
2
*
4
7
) =
-
1
7
;
a
0
3,5
=
0
- (
2
*
-
1
7
) =
2
7
;
a
0
3,6
=
0
- (
2
*
2
7
) =
-
4
7
;
a
0
3,7
=
1
- (
2
*
0
) =
1
;
a
0
3,8
=
0
- (
2
*
0
) =
0
;
a
0
4,2
=
-1
1
2
- (
-1
1
2
*
1
) =
0
;
a
0
4,3
=
-3
1
2
- (
-1
1
2
*
-
3
7
) =
-4
1
7
;
a
0
4,4
=
4
- (
-1
1
2
*
4
7
) =
4
6
7
;
a
0
4,5
=
-1
1
2
- (
-1
1
2
*
-
1
7
) =
-1
5
7
;
a
0
4,6
=
0
- (
-1
1
2
*
2
7
) =
3
7
;
a
0
4,7
=
0
- (
-1
1
2
*
0
) =
0
;
a
0
4,8
=
1
- (
-1
1
2
*
0
) =
1
;
Hide description
5
Iteration 3Divide
3
th row by
5
6
7
;
1
0
0
0
1
2
1
0
0
1
1
2
-
3
7
1
-4
1
7
0
4
7
-
1
41
4
6
7
1
2
-
1
7
2
41
-1
5
7
0
2
7
-
4
41
3
7
0
0
7
41
0
0
0
0
1
a
0
3,3
=
5
6
7
5
6
7
=
1
;
a
0
3,4
=
-
1
7
5
6
7
=
-
1
41
;
a
0
3,5
=
2
7
5
6
7
=
2
41
;
a
0
3,6
=
-
4
7
5
6
7
=
-
4
41
;
a
0
3,7
=
1
5
6
7
=
7
41
;
a
0
3,8
=
0
5
6
7
=
0
;
Hide description
From
4
th row we subtract
3
th row, multiplied by
-4
1
7
;
1
0
0
0
1
2
1
0
0
1
1
2
-
3
7
1
0
0
4
7
-
1
41
4
31
41
1
2
-
1
7
2
41
-1
21
41
0
2
7
-
4
41
1
41
0
0
7
41
29
41
0
0
0
1
a
0
4,3
=
-4
1
7
- (
-4
1
7
*
1
) =
0
;
a
0
4,4
=
4
6
7
- (
-4
1
7
*
-
1
41
) =
4
31
41
;
a
0
4,5
=
-1
5
7
- (
-4
1
7
*
2
41
) =
-1
21
41
;
a
0
4,6
=
3
7
- (
-4
1
7
*
-
4
41
) =
1
41
;
a
0
4,7
=
0
- (
-4
1
7
*
7
41
) =
29
41
;
a
0
4,8
=
1
- (
-4
1
7
*
0
) =
1
;
Hide description
6
Iteration 4Divide
4
th row by
4
31
41
;
1
0
0
0
1
2
1
0
0
1
1
2
-
3
7
1
0
0
4
7
-
1
41
1
1
2
-
1
7
2
41
-
62
195
0
2
7
-
4
41
1
195
0
0
7
41
29
195
0
0
0
41
195
a
0
4,4
=
4
31
41
4
31
41
=
1
;
a
0
4,5
=
-1
21
41
4
31
41
=
-
62
195
;
a
0
4,6
=
1
41
4
31
41
=
1
195
;
a
0
4,7
=
29
41
4
31
41
=
29
195
;
a
0
4,8
=
1
4
31
41
=
41
195
;
Hide description
7
Iteration 1From
3
th row we subtract
4
th row, multiplied by
-
1
41
;
From
2
th row we subtract
4
th row, multiplied by
4
7
;
1
0
0
0
1
2
1
0
0
1
1
2
-
3
7
1
0
0
0
0
1
1
2
53
1365
8
195
-
62
195
0
97
342
-
19
195
1
195
0
-
58
683
34
195
29
195
0
-
82
683
1
195
41
195
a
0
3,4
=
-
1
41
- (
-
1
41
*
1
) =
0
;
a
0
3,5
=
2
41
- (
-
1
41
*
-
62
195
) =
8
195
;
a
0
3,6
=
-
4
41
- (
-
1
41
*
1
195
) =
-
19
195
;
a
0
3,7
=
7
41
- (
-
1
41
*
29
195
) =
34
195
;
a
0
3,8
=
0
- (
-
1
41
*
41
195
) =
1
195
;
a
0
2,4
=
4
7
- (
4
7
*
1
) =
0
;
a
0
2,5
=
-
1
7
- (
4
7
*
-
62
195
) =
53
1365
;
a
0
2,6
=
2
7
- (
4
7
*
1
195
) =
97
342
;
a
0
2,7
=
0
- (
4
7
*
29
195
) =
-
58
683
;
a
0
2,8
=
0
- (
4
7
*
41
195
) =
-
82
683
;
Hide description
8
Iteration 2From
2
th row we subtract
3
th row, multiplied by
-
3
7
;
From
1
th row we subtract
3
th row, multiplied by
1
1
2
;
1
0
0
0
1
2
1
0
0
0
0
1
0
0
0
0
1
57
130
11
195
8
195
-
62
195
19
130
47
195
-
19
195
1
195
-
17
65
-
2
195
34
195
29
195
-
1
130
-
23
195
1
195
41
195
a
0
2,3
=
-
3
7
- (
-
3
7
*
1
) =
0
;
a
0
2,4
=
0
- (
-
3
7
*
0
) =
0
;
a
0
2,5
=
53
1365
- (
-
3
7
*
8
195
) =
11
195
;
a
0
2,6
=
97
342
- (
-
3
7
*
-
19
195
) =
47
195
;
a
0
2,7
=
-
58
683
- (
-
3
7
*
34
195
) =
-
2
195
;
a
0
2,8
=
-
82
683
- (
-
3
7
*
1
195
) =
-
23
195
;
a
0
1,3
=
1
1
2
- (
1
1
2
*
1
) =
0
;
a
0
1,4
=
0
- (
1
1
2
*
0
) =
0
;
a
0
1,5
=
1
2
- (
1
1
2
*
8
195
) =
57
130
;
a
0
1,6
=
0
- (
1
1
2
*
-
19
195
) =
19
130
;
a
0
1,7
=
0
- (
1
1
2
*
34
195
) =
-
17
65
;
a
0
1,8
=
0
- (
1
1
2
*
1
195
) =
-
1
130
;
Hide description
9
Iteration 3From
1
th row we subtract
2
th row, multiplied by
1
2
;
1
0
0
0
0
1
0
0
0
0
1
0
0
0
0
1
16
39
11
195
8
195
-
62
195
1
39
47
195
-
19
195
1
195
-
10
39
-
2
195
34
195
29
195
2
39
-
23
195
1
195
41
195
a
0
1,2
=
1
2
- (
1
2
*
1
) =
0
;
a
0
1,3
=
0
- (
1
2
*
0
) =
0
;
a
0
1,4
=
0
- (
1
2
*
0
) =
0
;
a
0
1,5
=
57
130
- (
1
2
*
11
195
) =
16
39
;
a
0
1,6
=
19
130
- (
1
2
*
47
195
) =
1
39
;
a
0
1,7
=
-
17
65
- (
1
2
*
-
2
195
) =
-
10
39
;
a
0
1,8
=
-
1
130
- (
1
2
*
-
23
195
) =
2
39
;
Hide description
Answer
B = A⁻¹16
39
11
195
8
195
-
62
195
1
39
47
195
-
19
195
1
195
-
10
39
-
2
195
34
195
29
195
2
39
-
23
195
1
195
41
195
SIZE4×4METHODGaussian elimination