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Inverse matrix

Definition.
Inverse matrix A−1 is the matrix, the product of which to original matrix A is equal to the identity matrix I:

A·A-1 = A-1·A = I

Note.
The inverse matrix exists only for square matrices whose determinant is not equal to zero.

Inverse matrix - properties

det(A-1) =  1
det(A)
(A·B)-1 = A-1·B-1
(A-1)T = (AT)-1
(kA)-1 A-1
k
(A-1)-1 = A

Inverse matrix - methods of calculation

Use Gaussian elimination to calculate inverse matrix

  • Adjoin the identity matrix onto the right of the original matrix, so that you have A on the left side and the identity matrix on the right side. It will look like this [ A | I ].
  • Row-reduce the matrix until the left side to the Identity matrix. When the left side is the Identity matrix, the right side will be the Inverse [ I | A-1 ]. If you are unable to obtain the identity matrix on the left side, then the matrix is singular and has no inverse.
  • Take the augmented matrix from the right side and call that the inverse
Example 1.
Find the inverse matrix of the matrix A
A = (241)
021
211

Solution: Adjoin the identity matrix onto the right of the matrix A:

A|E = (241100) ~
021010
211001

Row-reduce the matrix until the left side to the Identity matrix.
R3 - R1 → R3 (multiply 1 row by -1 and add it to 3 row):

(241100) ~ 
021010
2 - 21 - 41 - 10 - 10 - 01 - 0

 ~ (241100) ~
021010
0-30-101

R3 / 3 → R3 (divide 3 row by -3):

(241100) ~ 
021010
0101/30-1/3

R2 ↔ R3 (Interchange the 3 and 2 rows):

 ~ (241100) ~
0101/30-1/3
021010

R1 - 4 R2 → R1 (multiply 2 row by -4 and add it to 1 row); R3 - 2 R2 → R3 (multiply 2 row by -2 and add it to 3 row):

(2 - 4·04 - 4·11 - 4·01 - 4·(1/3)0 - 4·00 - 4·(-1/3)) ~ 
0101/30-1/3
0 - 2·02 - 2·11 - 2·00 - 2·1/31 - 2·00 - 2·(-1/3)

 ~ (201-1/304/3) ~
0101/30-1/3
001-2/312/3

R1 - R3 → R1 (multiply 3 row by -1 and add it to 1 row):

(2 - 00 - 01 - 1-1/3 - (-2/3)0 - 14/3 - 2/3) ~ 
0101/30-1/3
001-2/312/3

 ~ (2001/3-12/3) ~
0101/30-1/3
001-2/312/3

R1 / 2 → R1 (divide 1 row by 2):

(1001/6-1/21/3)
0101/30-1/3
001-2/312/3

Answer: A-1(1/6-1/21/3)
1/30-1/3
-2/312/3

Use matrix of cofactors to calculate inverse matrix

Definition.
Matrix C, elements of which are the cofactors of the corresponding elements of the matrix A is called the matrix of cofactors.
Definition.
Transpose of the matrix of cofactors, known as an adjugate matrix.
A-1 1 CT
det(A)
Example 1.
Find the inverse matrix of the matrix A
A = (241)
021
211

Solution: Find the determinant of matrix A:

det(A) =  241  = 
021
211

= 2·2·1 + 4·1·2 + 1·0·1 - 1·2·2 - 2·1·1 - 4·0·1 = 4 + 8 + 0 - 4 - 2 - 0 = 6

Find the matrix of cofactors C:

C11 = (-1)1 + 1· 2 1  = 2·1 - 1·1 = 1
1 1

C12 = (-1)1 + 2· 0 1  = -(0·1 - 1·2) = 2
2 1

C13 = (-1)1 + 3· 0 2  = 0·1 - 2·2 = -4
2 1

C21 = (-1)2 + 1· 4 1  = -(4·1 - 1·1) = -3
1 1

C22 = (-1)2 + 2· 2 1  = 2·1 - 1·2 = 0
2 1

C23 = (-1)2 + 3· 2 4  = -(2·1 - 4·2) = 6
2 1

C31 = (-1)3 + 1· 4 1  = 4·1 - 1·2 = 2
2 1

C32 = (-1)3 + 2· 2 1  = -(2·1 - 1·0) = -2
0 1

C33 = (-1)3 + 3· 2 4  = 2·2 - 4·0 = 4
0 2

Write matrix of cofactors:

C = (12-4)
-306
2-24

Find the inverse matrix:

A-1 1 CT  =  1
det(A) 6
(1-32)
20-2
-464
 = 
(1/6-1/21/3)
1/30-1/3
-2/312/3

Answer: A-1(1/6-1/21/3)
1/30-1/3
-2/312/3

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