**Operation of Matrices**

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

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**Addition of Matrices**

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 the same as the order of
the matrix A.

**Algebraic Properties of Matrix Addition**

The 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)

**4. ****Existence**** of additive identity:**

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

**5. ****Existence**** of additive
inverse:**

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*

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