Properties of Matrix Multiplication

1. Powers of a Matrix
   If matrix is a square matrix and and are natural numbers:
1. , , ,
2. ,


2. Properties of Matrix Multiplication
   In matrix multiplication, the associative and distributive properties hold, but the commutative property does not hold. For three matrices , , and where both addition and multiplication are defined:
1. (the commutative property does not hold)
2. (associative property)
3. , (distributive property)
4. (where is a real number)
5. (where is the zero matrix)


3. Identity Matrix
1. An square matrix where the diagonal elements and all other elements are zero is called an identity matrix of order , denoted by .
   
   
   
2. For an matrix and the identity matrix , the following holds:
3. Properties of the Identity Matrix
1. , , , (where is a natural number)
2. , ,