Why is matrix multiplication useful? All of the trouble I have in learning linear algebra seems to come from my lack of understanding of its applications.  I understand how to do the multiplication, and I understand that it is a necessary tool of matrix algebra, but I want to know what the multiplication actually does, or what does its result actually represent?  
Any modern real world examples that are relatively simple and specific would help me immensely.
Thanks.

Linear Algebra is often helpful when you have multiple equations with multiple unknowns. Say you have $20$ equations with $20$ unknowns. Could you sit down and keep substituting equations into each other and get an answer? Yes. Is it time effective? Absolutely not.
Linear Algebra allows you to represent these equations using matrices to help solve for unknowns. The most common form $Ax = b$ is a prime example of that. You can just take the inverse of $A$ if $A$ is nonsingular to solve for your variable matrix $x$.
$$Ax = b$$
$$A^{-1}Ax = A^{-1}b$$
$$x = A^{-1}b$$
A real world example would be say you have $6$ ropes all attached to a block pulling in different directions. The tension on the block could be represented by $6$ equations with the angles at which the ropes are pulling on it. The tension in each rope could easily be solved by using matrix algebra.
