Just like Normal arithmetics, Simplification of Matrices is governed by similar rules, yet it's very important to know the basic properties of matrices

Properties of Matrix Operations

Properties of Addition

The basic properties of addition for real numbers also hold true for matrices. 

Let A, B and C be m x n matrices

A + B  =  B + A    commutative
A + (B + C)  =  (A + B) + C    associative

There is a unique m x n matrix O with

A + O  =  A        additive identity

For any  m x n matrix A there is an m x n matrix B (called -A) with

A + B  =  O        additive inverse

 

The proofs are all similar.  We will prove the first property.

Properties of Matrix Multiplication

Unlike matrix addition, the properties of multiplication of real numbers do not all generalize to matrices.  Matrices rarely commute even if AB and BA are both defined.  There often is no multiplicative inverse of a matrix, even if the matrix is a square matrix.  There are a few properties of multiplication of real numbers that generalize to matrices.  We state them now.

Let A, B and C be matrices of dimensions such that the following are defined.  Then

A(BC)  =  (AB)C                 associative

A(B + C)  =  AB + AC        distributive

(A + B)C  =  AC + BC        distributive

There are unique matrices I with

 IA  =  AI  =  A     multiplicative identity