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# Operations on Matrices - Properties of Matrix Addition

#### description

• Commutative Law
• Associative Law
• The existence of additive inverse

#### notes

Commutative Law:
If A = [a_(ij)], B = [b_(ij)] are matrices of the same order, say m × n, then A + B = B + A.
Now A + B
= [a_(ij)] + [b_(ij)]  = [a_(ij) + b_(ij)]
= [b_(ij) + a_(ij)] (addition of numbers is commutative) = ([b_(ij)] + [a_(ij)])= B + A

Associative Law:
For any three matrices A = [a_(ij)], B = [b_(ij)], C = [c_(ij)] of the same order, say m × n, (A + B) + C = A + (B + C).
Now (A + B) + C = ([a_(ij)] + [b_(ij)]) + [c_(ij)]
= [a_(ij) + b_(ij)] + [c_(ij)] = [(a_(ij) + b_(ij)) + c_(ij)]
= [a_(ij) + (b_(ij) + c_(ij))] (Why?)
= [a_(ij)] + [(b_(ij) + c_(ij))] = [a_(ij)] + ([b_(ij)] + [c_(ij)]) = A + (B + C)

Existence of additive identity:  Let A = [aij] be an m × n matrix and O be an m × n zero matrix, then A + O = O  + A = A. In other words, O is the additive identity for matrix addition.

Let A = [a_(ij)]_(m × n) be any matrix, then we have another matrix as
– A = [– a_(ij)]_(m × n) such that
A + (– A) = (– A) + A= O.
So – A is the additive inverse of A or negative of A.

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Matrices part 11 (Property of matrices Addition) [00:12:55]
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