3.1 INTRODUCTION

The n ^th order set of simultaneous, linear algebraic equations in the n unknowns, x ₁,x ₂,x ₃,...,x _n,

can conveniently be represented by the matrix equation

or more simply by

where A, as indicated in Section 1.2, is a square matrix of coefficients having elements a _ij and where X and Y are n 1 column vectors with elements x and y, respectively.

Because division of matrices is not permitted, one method for the solution of matrix equations such as the one shown in eq (3.1) is called matrix inversion.

3.2 MATRIX INVERSION

If eq (3.1) is premultiplied by an n n square matrix B so that BAX = BY, a solution for the unknowns, X, will evolve if the product BA is equal to the identity matrix I

or

If

the matrix B is said to be the inverse of A

and, of course, the inverse of the inverse is the matrix itself

or

It may be recalled that, in general, matrix multiplication is not commutative. The multiplication of a matrix by its inverse is one specific case where matrix multiplication is commutative

3.3 PROPERTIES OF THE INVERSE

The inverse of a product of two matrices is the product of the inverses taken in reverse order. This is easily proved. Consider the product AB and postmultiply by B ^?1A. Because...