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
or
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.
Table of Contents
Invers of 2×2 Matrix
Remember about identity of 2×2 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 3×3 Matrix
Remember about identity of 3×3 matrix.
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.
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
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.
2. Determine the invers of matrix N.
Using minor-cofactor method (it depends on your choice)
3. Determine the value of x if matrix M is singular matrix.
Singular matrix means matrix M has no invers. In another word the determinant is zero.
Then
|M| = 2(-8) – 4.x = 0
-16 – 4x = 0
-16 = 4x
x = -4