` ` Matrix Definition in Math**Introduction**:

While solving linear systems of equations, a new notation was introduced to reduce the amount of writing. For this new notation, the word matrix was first used by the English mathematician **James Sylvester (1814 – 1897). Arthur Cayley (1821 – 1895) **developed the theory of matrices and used them in linear transformations. Nowadays, matrices are

used in high-speed computers and also in other various disciplines.

The concept of determinants was used by Chinese and Japanese but the Japanese mathematician **Seki Kowa (1642 – 1708)** and the German mathematician Gottfried Wilhelm Leibniz (1646 – 1716) is credited for the invention of determinants. **G. Cramer (1704 – 1752)** applied the determinants successfully for solving the systems of linear equations.

**Matrix definition**:** A rectangular array of numbers enclosed by a pair of brackets such as:**

**is called matrix.**

The horizontal lines of numbers are called rows and the vertical lines of numbers are called columns. The numbers used in rows or columns are said to be the **entries or elements of the matrix.**

The matrix in **(i)** has two rows and three columns while the matrix in **(ii)** has 4 rows and three columns. Note that the number of elements of the matrix in (ii) is [katex]4\times 3= 12[/katex]. Now we give a general definition of a matrix.

Generally, a bracketed rectangular array of [katex]m\times n[/katex] elements

[katex]aij (i = 1, 2, 3, …., m; j = 1, 2, 3, …., n)[/katex], arranged in [katex]m[/katex] rows and [katex]n[/katex] columns such as:

is called an m by n matrix (written as [katex]m\times n[/katex] matrix).

[katex]m\times n[/katex] is called the order of the matrix in **(iii)** .

We usually use capital letters such as A, B, C, X, Y, etc., to represent the matrices and small letters such as a, b, c,…, m, n,…, [katex]a_{11}, a_{12}, a_{13}[/katex] …., etc., to indicate the entries of the matrices.

Let the matrix in (iii) be denoted by A. The ith row and the jth column of A are indicated

in the following tabular representation of A.