Introduction

Before diving into formulas, it helps to see where matrices show up in everyday life. Think of the precise rows and columns of soldiers in a Republic Day parade, students seated by row and column in a class photograph, or seats numbered by row and column in a cinema hall — in every case, arranging items into rows and columns makes it easy to locate any single element using just two numbers: which row, and which column. A matrix formalises this idea. It is a rectangular arrangement of numbers, symbols or expressions into rows and columns, and each individual entry in that arrangement is called an element of the matrix. A matrix is usually named with a capital letter, and its general element is written $a_{ij}$ — the entry sitting in the $i$-th row and $j$-th column, so a matrix $A$ is compactly written as $A = [a_{ij}]$, where $1 \le i \le m$ and $1 \le j \le n$. This row-and-column …