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# Properties of Matrix Addition, Scalar Multiplication and Product of Matrices

Let A, B, and C be three matrices of same order which are conformable for addition and a, b be two scalars.

Properties of Matrix Addition, Scalar Multiplication and Product of Matrices

Let A, B, and C be three matrices of same order which are conformable for addition and a, b be two scalars. Then we have the following:

(1) A + B yields a matrix of the same order

(2) A + B = B + A (Matrix addition is commutative)

(3) (A + B) + C = A + (B + C) (Matrix addition is associative)

(4) A + O = O + A = A (O is additive identity)

(5) A + (- A)=O = (- A) + A (- A is the additive inverse of A)

(6) (a + b)A = aA + bA and a(A + B) = aA + aB

(7) a(bA) =(ab)A,  1A = A and 0A = O.

## Properties of matrix multiplication

Using the algebraic properties of matrices we have,

• If A, B, and C are three matrices of orders m × n, n × p and p × q respectively, then A(BC) and (AB)C are matrices of same order m × q and

A(BC) = (AB)C (Matrix multiplication is associative).

•If A, B, and C are three matrices of orders m × n, n × p, and n × p respectively, then A(B + C) and AB + AC are matrices of the same order m × p and      • A(B + C) = AB + AC. (Matrix multiplication is left distributive over addition)

If A, B, and C are three matrices of orders m × n, m × n, and n × p respectively, then (A + B)C and AC + BC are matrices of the same order m × p and

(A + B)C = AC + BC. (Matrix multiplication is right distributive over addition).

• If A, B are two matrices of orders m × n and n × p respectively and α is scalar, then α(AB) = A(α B) = (α A)B is a matrix of order m × p.

If I is the unit matrix, then AI = IA = A (I is called multiplicative identity).

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