Linear Algebra 2- Properties of Matrix Multiplication (2.1)

Matrix Multiplication: 
Theorem 2: Let A be an m x m matrix, and let B and C have sizes for which the indicated sums and products are defined.
a.) A(BC) = (AB)C                                                (Associative Law of Multiplication)
corresponds to composition of linear transformations.
b.) A(B+C) = AB+AC                                           (Left Distributive Law)
c.) (B+C)A = BA+CA                                           (Right Distributive Law)
d.) r(AB) = (rA)B = A(rB)
for any scalar r.
e.) I_mA = A = AI_m                                             (Identity for Matrix Multiplication)

Example: Product 

Example: Associative Law of Multiplication: 

Example: Identity for Matrix Multiplication:

Warnings: 
1. In general, AB does not equal BA.
2. The cancellation laws do not fold for matrix multiplication. that is, if AB = AC, then it is not true in general that B = C.
3. If a product AB is the zero matrix, you cannot conclude in general that either A = 0 or B = 0.

Powers of a Matrix: 
If A is an n x n matrix and if k is a positive integer, then A^k denotes the product of k copies of A.
                                                    A^k = A x A x A.....
If A is nonzero and if x is in R^n, then A^k is the result of left multiplying x by A in the amount of k times. If k = 0 then A^0x should be x itself, which will be the identity matrix.

Example: 

 











Example: 

If you still need some help with matrix multiplication here is a good link to help you out.
https://www.mathsisfun.com/algebra/matrix-multiplying.html
Also i have found a really good video on Powers of a Matrix and how to use the powers as an implication of multiply matrix's. Its called power of Matrix's and you can find it on Youtube.

An example problem could be: Find all properties from theorem 2 (a-e) and also find the power matrix's for A, B, and C.