Matrices

A matrix is a rectangular array of entries — arranged in rows and columns and enclosed in square brackets $[\ ]$ — usually named with capital letters $A, B, C, \ldots$. In general, a matrix's entries may be real numbers, complex numbers, or even functions of one or more variables; in this chapter, entries are restricted to real numbers or real-valued functions of a real variable.

If a matrix $A$ has $m$ rows and $n$ columns, it is written

where $a_{ij}$ — the $(i,j)$th element — is the entry sitting in row $i$ and column $j$. The pair $m\times n$ (read "$m$ by $n$") is the order (or size) of $A$; $m$ and $n$ are positive integers. The $i$th row of $A$ is itself a $1\times n$ matrix, and the $j$th column is an $m \times 1$ matrix.

Worked illustration. A student's marks across three exams (rows) in five subjects — Tamil, English, Mathematics, Science, Social Science (columns) — can be packaged as a single $3\times 5$ matrix

The $(3,2)$ entry, $84$, is the mark scored in English in Exam 3 — reading off any single fact from the table is just reading off one entry $a_{ij}$ of the matrix. …