#### 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.