History of the Matrix

The matrix has a long history of application in solving linear equations. They were known as arrays until the $1800$'s.  The term "matrix" (Latin for "womb", derived from mater—mother) was coined by James Joseph Sylvester in $1850$, who understood a matrix as an object giving rise to a number of determinants today called minors, that is to say, determinants of smaller matrices that are derived from the original one by removing columns and rows.  An English mathematician named Cullis was the first to use modern bracket notation for matrices in $1913$ and he simultaneously demonstrated the first significant use of the notation $A=a_{i,j}$ to represent a matrix where $a_{i,j}$ refers to the element found in the ith row and the jth column.  Matrices can be used to compactly write and work with multiple linear equations, referred to as a system of linear equations, simultaneously. Matrices and matrix multiplication reveal their essential features when related to linear transformations, also known as linear maps.

What is a Matrix 

In mathematics, a matrix (plural matrices) is a rectangular array of numbers, symbols, or expressions, arranged in rows and columns. Matrices are commonly written in box brackets. The horizontal and vertical lines of entries in a matrix are called rows and columns, respectively. The size of a matrix is defined by the number of rows and columns that it contains. A matrix with m rows and n columns is called an m × n matrix or $m$-by-$n$ matrix, while m and n are called its dimensions. The dimensions of the following matrix are $2 \times 3$ up(read "two by three"), because there are two rows and three columns.

$A={\displaystyle {\begin{bmatrix}1&9&-13\\20&5&-6\end{bmatrix}}}$

Matrix Dimensions

Each element of a matrix is often denoted by a variable with two subscripts. For instance, $a_{2,1}$ represents the element at the second row and first column of a matrix A.

The individual items (numbers, symbols or expressions) in a matrix are called its elements or entries.   Provided that they are the same size (have the same number of rows and the same number of columns), two matrices can be added or subtracted element by element. The rule for matrix multiplication, however, is that two matrices can be multiplied only when the number of columns in the first equals the number of rows in the second. Any matrix can be multiplied element-wise by a scalar from its associated field.

Matrices which have a single row are called row vectors, and those which have a single column are called column vectors. A matrix which has the same number of rows and columns is called a square matrix. In some contexts, such as computer algebra programs, it is useful to consider a matrix with no rows or no columns, called an empty matrix.