Inverse matrix

• Inverse matrix - definition
• Inverse matrix - properties
• Inverse matrix - methods of calculation
  • Gaussian elimination
  • Matrix of cofactors (adjugate matrix)

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

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

• 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

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

Matrix C, elements of which are the cofactors of the corresponding elements of the matrix A is called the matrix of cofactors.

Transpose of the matrix of cofactors, known as an adjugate matrix.

A-1 =	1	CT
	det(A)

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