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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