ALGEBRAIC PROPERTIES OF MATRICES

Properties of Matrix Addition and Scalar Multiplication

Let A, B, C be m x n matrices and p and q be two non-zero scalars (numbers). Then we have the following properties.

Commutative Property of Matrix Addition : 

A + B = B + A

Associative Property of Matrix Addition :

A + (B + C) = (A + B) + C

Associative Property of Scalar Multiplication :

(pq)A = p(qA)

Scalar Identity Property : 

AI = IA = A

where I is the unit matrix.

Distributive Property of Scalar and Two Matrices : 

p(A + B) = pA+ pB

Distributive Property of Two Scalars with a Matrix :

(p + q)A = pA + qA

Additive Identity

The null matrix or zero matrix is the identity for matrix addition.

Let A be any matrix.

Then,

A + 0 = 0 + A = A

where 0 is the null matrix or zero matrix of same order as that of A.

Additive Inverse

If A be any given matrix then –A is the additive inverse of A.

In fact we have

A +(-A) = (-A) + A = 0

Properties of Multiplication of Matrix

(a) Matrix multiplication is not commutative in general :

If A is of order m x n and B of the order n x p then AB is defined but BA is not defined. Even if AB and BA are both defined, it is not necessary that they are equal.

In general AB ≠ BA.

(b) Matrix multiplication is distributive over matrix addition :

(i) If A, B, C are m x n , n x p and n x p matrices respectively then

A(B + C) = AB + AC (Right Distributive Property)

(ii) If A, B, C are m x n, m x n and n x p matrices respectively then

(A + B)C = AC + BC (Left Distributive Property)

(c) Matrix multiplication is always associative :

If A, B, C are m x n , n x p and p x q matrices respectively then

(AB)C = A(BC)

(d) Multiplication of a matrix by a unit matrix :

If A is a square matrix of order n x n and I is the unit matrix of same order then

AI = IA = A

(I is called multiplicative identity)

Note : 

If x and y are two real numbers such that xy = 0 then either x = 0 or y = 0 . But this condition may not be true with respect to two matrices.

AB = 0 does not necessarily imply that A = 0 or B = 0 or both A,B = 0

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