Introduction to Matrices

4.4 Introduction to Matrices

The theory of matrices was developed by the mathematician Arthur Cayley. Matrices express numerical information compactly and are used to represent operators — they are essential in Economics, Statistics and Computer Science.

Definition. A rectangular arrangement of $mn$ numbers in $m$ rows and $n$ columns, enclosed in $[\ ]$ or $(\ )$, is called a matrix of order $m\times n$ (read "$m$ by $n$"). A matrix by itself does not have a single value or special meaning — this is the key difference from a determinant, which IS a single number.

Each entry is an element of the matrix. Matrices are usually named $A,B,C,\dots$ and their elements $a_{ij},b_{ij},\dots$, where $a_{ij}$ is the element in row $i$, column $j$. In general,

Illustrative examples

$A=\begin{bmatrix}2&-3&9\\1&0&-7\\4&-2&1\end{bmatrix}$ is $3\times3$ (9 elements); here $a_{32}=-2$. …