Matrix is a rectangular array of ordered set of numbers which are arranged or organized in rows and columns in a table. The Rows are the horizontal lines and the Columns are the vertical lines. Each numbers in the matrix are called its elements or its entries.
The matrix below say A has m rows and n columns, where m and n are called its dimensions. Then the matrix with m rows and n columns is called m-by-n matrix or $m\times n$ matrix. The entry of a matrix A in the ith row and jth column is called the ij or (ij)th entry of A. The (ij)th entry of a matrix A is written as aij or A[ij].

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Example
                  Matrix Example
This matrix has $m = 4$ rows, and $n = 3$ columns.
The marix $m\times n$ is $4\times 3$ matrix.
Element a32 or A[32] is 1.
Matrix Addition:
Let A and B are two matrices of same type $m \times n$. Then their sum is defined to be the matrix of the type $m \times n$ obtained by adding the corresponding elements of A and B.
The sum of two m-by-n matrices say A and B is calculated as:

Addition Matrix

$(A + B)_{ij} = A_{ij} + B_{ij}, where  1\leq i \leq m   and  1\leq j \leq n$

Matrix Subtraction:
Matrix subtraction is calculated just like matrix addition but here we are subtracting the two matrices.
Subtraction Matrix$(A - B)_{ij} = A_{ij} - B_{ij}, where  1\leq i \leq m   and  1\leq j \leq n$
In the case of matrix addition and subtraction it should be noted that addition and subtraction is defined only for matrices which are of same size. 
Matrix Multiplication:
If A is a $n\times m$ matrix and B is $m\times p$ matrix,the product AB is $n\times p$ matrix.
Multiplication Matrix
[A B]$_{ij}$ = $ \sum_{k=1}^{m}A_{ik}B_{kJ}$

Matrix multiplication of two matrices is only possible if the number of column of first is equal to the number of rows of the second.