Matrices and Their Types

A matrix is a rectangular arrangement of numbers set out in horizontal rows and vertical columns and enclosed in brackets. Each number in the arrangement is an element (or entry) of the matrix. Matrices give business mathematics a compact language for handling many numbers at once — a price list across several products and outlets, the quantities in a set of orders, or the coefficients of a system of linear equations — and, as later sections show, they let a whole system of equations be solved in one stroke.

A matrix having $m$ rows and $n$ columns is said to be of order $m\times n$ (read “$m$ by $n$”), and a general element is written $a_{ij}$, where $i$ is its row number and $j$ its column number. For example, $A=\begin{pmatrix} 2 & -1 & 5 \\ 0 & 3 & 4 \end{pmatrix}$ is a matrix of order $2\times 3$; here $a_{11}=2$, $a_{13}=5$ and $a_{23}=4$. Note carefully that the row count is always stated first: an order $m\times n$ is not the same as $n\times m$.

The types of matrices met in this course are the following.

• Row matrix — a matrix with exactly one row, order $1\times n$, e.g. $\begin{pmatrix} 3 & -2 & 7 \end{pmatrix}$.
• Column matrix — a matrix with exactly one column, order $m\times 1$, e.g. $\begin{pmatrix} 4 \\ 1 \\ 6 \end{pmatrix}$.
• Square matrix — a matrix with as many rows as columns, order $n\times n$. Only a square matrix has a determinant or an inverse.
• Diagonal matrix — a square matrix in which every element off the leading (top-left to bottom-right) diagonal is zero, e.g. $\begin{pmatrix} 5 & 0 \\ 0 & -2 \end{pmatrix}$.
• Scalar matrix — a diagonal matrix whose diagonal elements are all equal, e.g. $\begin{pmatrix} 3 & 0 \\ 0 & 3 \end{pmatrix}$.
• Unit (identity) matrix — a scalar matrix whose common diagonal element is $1$, written $I$, e.g. $I_2=\begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}$. It plays the role of the number $1$: $AI=IA=A$.
• Zero (null) matrix — a matrix, of any order, all of whose elements are $0$, written $O$; it behaves like the number $0$ under addition.
• Upper / lower triangular matrix — a square matrix with all elements below (respectively above) the leading diagonal equal to $0$.

Two matrices are equal only when they have the same order and every pair of corresponding elements is equal. This is the idea behind many short problems: setting corresponding entries equal turns a matrix equation into ordinary equations in the unknown entries.