## Operation of Matrices

Here in Operation of Matrices, we deal with the Addition of Matrices, Subtraction of Matrices and Multiplication of a Matrix by a scalar (real number).

If A and B are two matrices of the same order, then A and B are said to be Conformable or Compatible for addition. The sum of A and B is denoted by A + B and it is obtained by adding corresponding elements of matrices A and B.

The matrix A + B will be of the same order as each of the matrices A and B is.

### Subtraction of Matrices

If A and B are two matrices of the same order, then they are said to be conformable for subtraction. The difference of the matrix B from A is denoted by A – B and it is obtained by subtracting the elements of B from the corresponding elements of A.

The order of the matrix A – B is same as the order of A or B.

### Multiplication of a Matrix by a scalar (real number)

If A is any matrix and k is any constant or a scalar, then the matrix obtained by multiplying each element of A by k is denoted by kA and it is called scalar multiple of A by k.

The order of the matrix kA is same as the order of the matrix A.

### Algebraic Properties of Matrix Addition

Addition of matrices satisfies the following properties:

1.   Closure property:

If A and B are two matrices of the same order, then their sum A + B is also a matrix of the same order as that of A or B.

2.   Commutative property:

If A and B are two matrices of the same order, then A + B = B + A.

3.   Associative properties:

If A, B and C are three matrices of the same order, then (A + B) + C = A + (B + C)

If A is any matrix, then there exists a null matrix O of the same order such as A + O = O + A = A.

If A is a matrix of any order, then there exists another matrix –A or same order such that A + (-A) = (-A) + A = O, the additive identity.

6.   If A and B are the matrices of the same order and k is a scalar, then k(A + B) = kA + kB.

7.   If A is a matrix and c, k are any two scalars, then (c + k)A = cA + kA.

8.   If c, k are any two scalars and A is a matrix, then c(kA) = (ck)A.

Worked Out Examples

You can comment your questions or problems regarding the addition, subtraction or scalar multiplication of matrices here.