Invers of Matrix: Gauss Jordan & Minor-Cofactor Methods, Examples

Invers of a matrix relate with identity of a matrix. It is symbolized by “X-1” where “X” is a matrix X. if there is matrix A, then the invers is A-1. The result of multiplication between matrix A and A-1 is identity matrix “I”. In mathematics symbol, it is written as

A.A-1 = I


A-1.A = I

But, not all of matrix has the invers. Besides that, even though it is a multiplication operation, but it doesn’t mean that identity matrix divide by invers of matrix A is matrix A.

Invers of 2×2 Matrix

Remember about identity of 2×2 matrix.

Invers of 2x2 Matrix

Invers of matrix X not only use determinant concept but also swap the a and d position and change b and c to be negative (multiplying with -1).

In another words, based on the definition,

X .X-1 = I

Invers of 2x2 Matrix b

Invers of 3×3 Matrix

Remember about identity of 3×3 matrix.

Identify Matrix 3x3

There are two methods to solve invers problem.

1. Gauss-Jordan Method

Gauss-Jordan method is simple but tricky. It is because the main concept is changing matrix to be identity matrix using simple operation (addition, subtraction, division, multiplication, and swap) using the row. If there is matrix A, then,

[A|I]  [I|A-1]

The process to solve invers called elementary row operations.

For example, there is matrix B3×3.

Invers of 3x3 Matrix Gauss-Jordan

Note: there is no absolute ways to get the invers. It depends on your creativity, trick, and carefulness.

2. Minor-Cofactor Method

Solving invers use minor-cofactor method is same concept with solving determinant problem. There are some steps in invers:

a. Determine the minor
b. Determine the matrix of cofactor
c. Determine the adjugate (Adjoint)
d. Determine the determinant
e. Determine the invers by multiplying 1/determinant with the adjoint

For example, if there is matrix C3×3 

Invers of 3x3 Matrix Minor-Cofactor

Impact of Invers Concept

There are other things as the impact of invers concept. Because of matrix has no dividing concept, then if there are matrix A and B and asked for matrix X,

AX = B  X = A-1.B

XA = B → X = B.A-1

Note: remember that matrix A.B ≠ B.A

There is no invers if …

Not all of matrices have invers. Some of them have no invers. The condition is if the determinant is zero, then there is no invers of the matrix. The matrix that has no invers called “singular matrix”.

Invers of Matrix Examples

1. Determine the invers of matrix M.

Invers of Matrix Example 1a
Invers of Matrix Example 1b

2. Determine the invers of matrix N.

Invers of Matrix Example 2a

Using minor-cofactor method (it depends on your choice)

Invers of Matrix Example 2b

3. Determine the value of x if matrix M is singular matrix.

Invers of Matrix Example 3

Singular matrix means matrix M has no invers. In another word the determinant is zero.


Invers of Matrix Example 3

|M| = 2(-8) – 4.x = 0

                -16 – 4x = 0

                        -16 = 4x

                            x = -4

Read Also

Matrix (Introduction)

Matrix Multiplication

Determinant of Matrix

Solving Systems of Linear Equations Using Matrix