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 ‚Č† 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.