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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 = 2 4 1 0 2 1 2 1 1

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

 A|E = 2 4 1 1 0 0 ~ 0 2 1 0 1 0 2 1 1 0 0 1

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):

 ~ 2 4 1 1 0 0 ~ 0 2 1 0 1 0 2 - 2 1 - 4 1 - 1 0 - 1 0 - 0 1 - 0

 ~ 2 4 1 1 0 0 ~ 0 2 1 0 1 0 0 -3 0 -1 0 1

R3 / 3 → R3 (divide 3 row by -3):

 ~ 2 4 1 1 0 0 ~ 0 2 1 0 1 0 0 1 0 1/3 0 -1/3

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

 ~ 2 4 1 1 0 0 ~ 0 1 0 1/3 0 -1/3 0 2 1 0 1 0

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 - 4·0 4 - 4·1 1 - 4·0 1 - 4·(1/3) 0 - 4·0 0 - 4·(-1/3) ~ 0 1 0 1/3 0 -1/3 0 - 2·0 2 - 2·1 1 - 2·0 0 - 2·1/3 1 - 2·0 0 - 2·(-1/3)

 ~ 2 0 1 -1/3 0 4/3 ~ 0 1 0 1/3 0 -1/3 0 0 1 -2/3 1 2/3

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

 ~ 2 - 0 0 - 0 1 - 1 -1/3 - (-2/3) 0 - 1 4/3 - 2/3 ~ 0 1 0 1/3 0 -1/3 0 0 1 -2/3 1 2/3

 ~ 2 0 0 1/3 -1 2/3 ~ 0 1 0 1/3 0 -1/3 0 0 1 -2/3 1 2/3

R1 / 2 → R1 (divide 1 row by 2):

 ~ 1 0 0 1/6 -1/2 1/3 0 1 0 1/3 0 -1/3 0 0 1 -2/3 1 2/3

 Answer: A-1 = 1/6 -1/2 1/3 1/3 0 -1/3 -2/3 1 2/3

### 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 = 2 4 1 0 2 1 2 1 1

Solution: Find the determinant of matrix A:

 det(A) = 2 4 1 = 0 2 1 2 1 1

= 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 = 1 2 -4 -3 0 6 2 -2 4

Find the inverse matrix:

 A-1 = 1 CT = 1 det(A) 6 1 -3 2 2 0 -2 -4 6 4
= 1/6 -1/2 1/3 1/3 0 -1/3 -2/3 1 2/3

 Answer: A-1 = 1/6 -1/2 1/3 1/3 0 -1/3 -2/3 1 2/3