What is the inverse of a matrix?

Let’s say A is an nxn square matrix and I is an identity matrix with dimension nxn. Let’s also assume B is another nxn square matrix, such that the following is true:

If such a matrix B exists for a square matrix A, then A is called an invertible matrix or a non-singular matrix. And the matrix B is called the inverse of the matrix A. The inverse of the square matrix A is denoted by A^-1.

So, we can say that, if A^-1 is the inverse of the square matrix A, then

Please note that a matrix A is invertible, if A has full rank (What is a full rank matrix?). In other words, a matrix A is invertible if all the columns and the rows are linearly independent. Moreover, for an invertible matrix A, the determinant of the matrix A should be non-zero.

Please note that an mxn matrix A has a left-inverse if there exists another matrix B such that:

And an mxn matrix A has a right inverse, if there exists another matrix B such that:

A square matrix A is invertible when there exists another matrix B such that:

How to calculate the inverse of a matrix using Python NumPy?

We can use the following Python code to calculate the inverse of a non-singular square matrix…