Humankind's first set of numbers was the set of natural numbers ( i.e., 1, 2, 3, ...). As time progressed, the number 0 was added to the set, forming whole numbers. This facilitated addition, but not subtraction, so with time, humans began thinking of both positive and negative numbers by using the set of all integers. This set allowed for subtraction and multiplication, but not division, and so the rational numbers were invented. Then people saw the need to express numbers such as the ratio between the circumference of a circle and its diameter (π) as well as numbers that could be multiplied by themselves to get a whole number or other positive numbers (i.e., square roots of positive numbers), and so the number system expanded to include the real numbers. As square roots of negative numbers became relevant, complex numbers were invented. In all these number systems, there are particular rules for operations such as multiplication and addition. Their respective operations were set up to define the newest sets of numbers (Allen, 1961).
In a similar way, matrices were invented to represent multiple values that are related to each other in some way. Just as the set of real numbers has its algebra to define multiplication and addition and their inverses, matrices have an algebra that determines how matrices can interact with each other. These rules were defined so that applying matrices to certain situations is useful.
As defined by Allen (1961), a matrix is "a rectangular array of entries appearing in rows and columns" (p. 14). It is built to contain large sets of related numbers as one single entity. Matrices are described using their order, which refers to the number of rows and columns. Generally, \(m\) is the number of rows and \(n\) is the number of columns in an \(m \times n\) matrix. When the matrix is square, meaning it has the same number of rows and columns, it is referred to as a square matrix of order \(n\), where \(n\) is the number of rows and columns. Let $$A= \begin{bmatrix} 1 & 0 \\ 2 & 1\\ \end{bmatrix}, B=\begin{bmatrix} 4 & -1 & 3 \\ 1 & 2 & 5\\ \end{bmatrix}. \quad $$ Then \(A\) is a square matrix of order \(2\) and \(B\) is a \(2 \times 3\) matrix.
Matrices can be used to describe points on a graph. For example, the matrix \(B\) above could be used to represent the points \((4,1)\), \((-1,2)\), and \((3,5)\) on the \(xy\)-coordinate plane. Similarly, a matrix with three rows (where \(m=3\)) can be used to represent points in the \(xyz\)-coordinate system.
Individual entries in a matrix are referred to by their row number (\(i\)) and column number (\(j\)). So an element without a specified row or column would be \(a_{ij}\). An element in the first row and second column would be referred to as \(a_{12}\). For example, the element \(a_{12}\) of matrix \(A\) is \(0\).
Matrices are defined as equal if they are of the same order and if each element in one is equivalent to the corresponding element in the other. That is, matrices \(C\) and \(D\) are equal if and only if the order of \(C\) equals the order of \(D\) and \(c_{ij}=d_{ij}\) for each \(i,j\) represented by the matrix.
The transpose of a matrix (\(B^t\) or \(B^T\)) is created by switching its rows and columns. For example, the transpose of matrix \(B\) above is $$B^t=\begin{bmatrix} 4 & 1 \\ -1 & 2\\ 3 & 5\\ \end{bmatrix} \quad.$$
These are some of the basic definitions associated with matrices, and some basic properties relating to these definitions are explored here.