Matrix

A matrix is a rectangular array of numbers, symbols or expressions arranged in rows and columns, enclosed in a single pair of brackets. If a matrix has $m$ rows and $n$ columns, we say it is of order $m \times n$ (read as $m$ by $n$). Every entry has an address: the element sitting in the $i$-th row and $j$-th column is written $a_{ij}$, so the whole matrix can be written compactly as

For example, a matrix of order $3 \times 2$ has $3$ rows, $2$ columns, and $3 \times 2 = 6$ elements in total. Within a matrix, the entries $a_{11}, a_{22}, a_{33}, \dots$ — where the row number equals the column number — are called the diagonal elements; every other entry, where the row and column numbers differ, is a non-diagonal element. This section builds the vocabulary — order, element, diagonal — that every later idea in this chapter, from special matrices to determinants and inverses, …