IAS Buddha - UPSC Preparation

The essence of Mathematics lies in its freedom. — Cantor

3.1 Introduction

A matrix is a compact way to write systems of linear equations, and is used in science, economics, cryptography, and computer graphics.

A matrix is an ordered rectangular array of numbers (or functions). Entries are called elements.

Order: A matrix with $m$ rows and $n$ columns has order $m \times n$ . It has $mn$ elements.

We write $A=[a_{ij}]_{m\times n}$ , where $a_{ij}$ is the entry in row $i$ and column $j$ .

Possible orders of a matrix with $8$ elements: $1\times 8$ , $8\times 1$ , $2\times 4$ , $4\times 2$ .

Column matrix: only one column, order $m\times 1$
Row matrix: only one row, order $1\times n$
Square matrix: $m=n$
Diagonal matrix: square, and $a_{ij}=0$ when $i\mathrel{\htmlClass{course-neq}{\char"2260}} j$
Scalar matrix: diagonal, and all diagonal entries equal a constant $k$
Identity matrix $I_n$ : $a_{ij}=1$ if $i=j$ , else $0$
Zero (null) matrix $O$ : every entry is $0$

Equality: $A=B$ if they have the same order and $a_{ij}=b_{ij}$ for all $i,j$ .

Multiplication: If $A$ is $m\times n$ and $B$ is $n\times p$ , then $AB=C$ is $m\times p$ with

$c_{ik}=\sum_{j=1}^{n} a_{ij}b_{jk}$

If $A=[a_{ij}]_{m\times n}$ , then $A'=[a_{ji}]_{n\times m}$ .

$A$ is symmetric if $A'=A$ .

$A$ is skew-symmetric if $A'=-A$ .

Every square matrix can be written as the sum of a symmetric matrix and a skew-symmetric matrix.

If $A$ and $B$ are symmetric of the same order, then $AB$ is symmetric if and only if $AB=BA$ .

If $AB=BA=I$ , then $B=A^{-1}$ .

The inverse of a square matrix, if it exists, is unique.