What is Matrix ?

Matrix

In mathematics, a matrix (plural: matrices) is a rectangular array of numbers, symbols, or expressions, arranged in rows and columns. For example, the dimensions of the matrix below are 2 × 3 (read "two by three"), because there are two rows and three columns:

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

Provided that they have the same size (each matrix has the same number of rows and the same number of columns as the other), two matrices can be added or subtracted element by element (see Conformable matrix). 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 (i.e., the inner dimensions are the same, n for an (m×n)-matrix times an (n×p)-matrix, resulting in an (m×p)-matrix. There is no product the other way round, a first hint that matrix multiplication is not commutative. Any matrix can be multiplied element-wise by a scalarfrom its associated field.

The individual items in an m×n matrix A, often denoted by a_i,j, where i and j usually vary from 1 to m and n, respectively, are called its elements or entries. For conveniently expressing an element of the results of matrix operations the indices of the element are often attached to the parenthesized or bracketed matrix expression; e.g.: (AB)_i,j refers to an element of a matrix product. In the context of abstract index notation this ambiguously refers also to the whole matrix product.