Inverse of a Non-Singular Square Matrix

Inverse of a Non-Singular Square Matrix

We recall that a square matrix is called a non-singular matrix if its determinant is not equal to zero and a square matrix is called singular if its determinant is zero. We have already learnt about multiplication of a matrix by a scalar, addition of two matrices, and multiplication of two matrices. But a rule could not be formulated to perform division of a matrix by another matrix since a matrix is just an arrangement of numbers and has no numerical value. When we say that, a matrix A is of order n, we mean that A is a square matrix having n rows and n columns.

In the case of a real number x ≠ 0, there exists a real number y (=1/x) called the inverse (or reciprocal) of x such that xy = yx = 1. In the same line of thinking, when a matrix A is given, we search for a matrix B such that the products AB and BA can be found and AB = BA = I , where I is a unit matrix.

In this section, we define the inverse of a non-singular square matrix and prove that a non-singular square matrix has a unique inverse. We will also study some of the properties of inverse matrix. For all these activities, we need a matrix called the adjoint of a square matrix.