
Addition and Subtraction of 2 Matrices
2 matrices and can be added to or subtracted from each other ONLY WHEN THEIR ORDER IS SAME. These operations are performed elementwise. e.g.
Then,
is always equal to , but may not always be equal to . We say matrix addition is commutative, but matrix subtraction is not commutative.

Multiplication of 2 Matrices
The multiplication operation is unique to the matrices in the way it is defined. It has immense applications in various fields. Please note how it is performed:
When is to be postmultiplied with , i.e. , the number of columns in must be equal to the number of rows in .
Let and .
Number of rows in = number of columns in . Hence, is possible and is given by:
Note that number of rows in is equal to the number of rows in and number of columns in is equal to the number of columns in .
It is not possible to premultiply by , i.e. , because number of columns in is NOT equal to number of rows in .

Elementary Transformations of a Matrix
A transformation transforms a matrix to another. There are 2 types of elementary transformations viz.

Row Transformations
Consider .
1)
Take th row. To each element of , add the number , such that both elements belong to same column. For example, would give,
Call this matrix .
2)
Take th row. Multiply each element of by a constant . For example, the matrix will get transformed to another matrix under .

Column Transformations
Again consider .
1)
Take th column. To each element of , add the number , such that both elements belong to same row. For example, would give,
Call this matrix .
2)
Take th column. Multiply each element of by a constant . For example, the matrix will get transformed to another matrix under .
NOTE: The transformations are a very useful tool in obtaining inverse of a matrix.