PUC Karnataka Science Class 12Department of Pre-University Education, Karnataka
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Operations on Matrices - Properties of Matrix Addition

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description

  • Commutative Law
  • Associative Law
  • Existence of additive identity
  • 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.

The existence of additive inverse:
 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.

Shaalaa.com | Matrices part 11 (Property of matrices Addition)

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