Inverse of a Matrix: Suppose A is an n x n matrix. The inverse of A is another n x n matrix which is denoted as A-1 and satisfies the following conditions.
AA-1 = A-1A = In
Below, with an example, we illustrate the relationship between a matrix and its inverse.
Does every Matrix have an inverse?
There are two ways to determine of square matrix has an inverse:
- For any given matrix with dimensions equal to n*n, if Rank of Matrix is not less than n then matrix has inverse. In other words, Rank of the square matrix must be equal to its dimension
- Non-Zero Determinant. If determinant of a square matrix is non zero then inverse of a matrix exits
How to Calculate Inverse of a Matrix: Let us suppose we have a matrix a given below.
Just do as mentioned: swap the positions of a and d, put negatives in front of b and c and divide everything by the determinant i.e. (ad-bc).
A square matrix that has an inverse is said to be non-singular or invertible; a square matrix that does not have an inverse is said to be singular.