Introduction

Introduction

A matrix of order $m \times m$ is a square arrangement of $m^2$ elements, and every such square matrix has a corresponding determinant -- the same elements, expanded out into a single numerical value. Standard XI already covered the types of matrices and the basic algebra of matrices: addition, subtraction, and multiplication of two matrices.

Matrices are not just an abstract bookkeeping tool -- they turn up in almost every branch of science. Many results in Statistics, Economics, and Operations Research are naturally expressed in matrix form, and matrices are sometimes even called the language of atomic Physics.

This chapter builds two further tools on top of that Class-XI foundation:

• Elementary transformations of a matrix (section 2.1) -- six standard row/column operations that convert a matrix into an equivalent one without changing what it represents.
• The inverse of a matrix (section 2.2) -- found either by elementary transformations (2.2.1) or by the adjoint method (2.2.2).

Both tools are needed to put matrices to real use, in particular for solving a system of linear equations (section 2.3), by the method of inversion or the method of reduction.