Intuition behind Matrix Multiplication If I multiply two numbers, say $3$ and $5$, I know it means add $3$ to itself $5$ times or add $5$ to itself $3$ times.  

But If I multiply two matrices, what does it mean ?  I mean I can't think it in terms of repetitive addition.  


  What is the intuitive way of thinking about multiplication of matrices?
Matrix ¨multiplication¨ is the composition of two linear functions. The composition of two linear functions is a linear function.

If a linear function is represented by A and another by B then AB is their composition. BA is the their reverse composition.

Thats one way of thinking of it. It explains why matrix multiplication is the way it is instead of piecewise multiplication.
This is the detailed and easiest explanation : https://nolaymanleftbehind.wordpress.com/2011/07/10/linear-algebra-what-matrices-actually-are/