Inverse Matrix Formula

If A and B are square matrices of the same frame, such that the product AB = BA = I, with I the identity matrix then B is the inverse matrix of A and vice versa, ie B = A^-1 or A = B^-1.

Inverse Matrix Formula

If A is a square matrix, then the inverse of the matrix A is:

Example:
Determine the inver of the matrix below!

Answer:
Determinant A(det(A)) is:

Minor from A is:
M₁₁ = | d | = d
M₁₂ = | c | = c
M₂₁ = | b | =  b
M₂₂ = | a | = a

The cofactor of A is:
C₁₁ = (-1)¹⁺¹ M₁₁ = d
C₁₂ = (-1)¹⁺² M₁₂ = c
C₂₁ = (-1)²⁺² M₂₁ = -b
C₂₂ = (-1)²⁺² M₂₂ = a

The cofactor matrix is:

The matrix adjoin is:

Then the inverse matrix becomes:

Note:
The matrix having an inverse is a matrix whose determinant value is ≠ 0, this matrix is called a nonsingular matrix, whereas the matrix whose value of determination = 0 is called a singular matrix.

Inverse a matrix if any and singular, then the properties apply:
(A^-1)^-1 = A
(A x B)^-1 = B^-1 x A^-1

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