AB multiplication = [A.col1B A.col2B ... A.colnB], so we can use linear combinations of the colums of A using the weights from the cols of B, or dot products of the rows of A with the cols of B.

For example, To obtain the ijth entry of AB we take the ith row of A and the jth column of B, and perform the dot product (line them up, multiply corresponding entries, and add).

Algebra of Matrices: A, B mxn matrices
1) A+B = B+A   commutative under +
2) A+(B+C)=(A+B)+C   associative under +
3) (cd)A=c(dA)
4) 1*A=A
5) c(A+B)=cA+cB
6) (c+d)A=cA+dA
        Define the zero matrix 0mn as additive identity
7) A+0mn=A
8) A+((-1)*A)=0mn -A=-1*A   add inverse of A
9) If cA=0mn then c=0 or A=0mn
10) A(BC)=(AB)C   associative under mult
11) A(B+C)=AB+AC   left distributive property
12) (A+B)C=AC+BC   right distributive property
13) c(AB)=(cA)B=A(cB)