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.

<br><br>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).
<p>
<font color=maroon>Algebra of Matrices:  A, B mxn matrices</font>
<br>1)  A+B = B+A &nbsp; <font color=maroon>commutative under +</font>
<br>2)  A+(B+C)=(A+B)+C &nbsp; <font color=maroon>associative under +</font>
<br>3) (cd)A=c(dA)

<br>4) 1*A=A
<br>5) c(A+B)=cA+cB
<br>6) (c+d)A=cA+dA
<br>&nbsp; &nbsp; &nbsp; &nbsp; 
<font color=maroon>Define the zero matrix 0mn as additive identity</font>
<br>7) A+0mn=A
<br>8) A+((-1)*A)=0mn  -A=-1*A &nbsp; 
<font color=maroon>add inverse of A</font>
<br>9) If cA=0mn then c=0 or A=0mn
<br>10) A(BC)=(AB)C &nbsp;
<font color=maroon>associative under mult</font>
<br>11) A(B+C)=AB+AC &nbsp;
<font color=maroon>left distributive property</font>
<br>12) (A+B)C=AC+BC &nbsp;
<font color=maroon>right distributive property</font>
<br>13) c(AB)=(cA)B=A(cB)