Question

How to do matrix multiplication

Answer 1

Matrix is an array of numbers:

 
A Matrix
(This one has 2 Rows and 3 Columns)

To multiply a matrix by a single number is easy:



These are the calculations:

2×4=82×0=02×1=22×-9=-18

We call the number ("2" in this case) a scalar, so this is called "scalar multiplication".

Multiplying a Matrix by Another Matrix

But to multiply a matrix by another matrix we need to do the "dot product" of rows and columns ... what does that mean? Let us see with an example:

To work out the answer for the 1st row and 1st column:



The "Dot Product" is where we multiply matching members, then sum up:

(1, 2, 3) • (7, 9, 11) = 1×7 + 2×9 + 3×11 
    = 58

We match the 1st members (1 and 7), multiply them, likewise for the 2nd members (2 and 9) and the 3rd members (3 and 11), and finally sum them up.

Want to see another example? Here it is for the 1st row and 2nd column:



(1, 2, 3) • (8, 10, 12) = 1×8 + 2×10 + 3×12 
    = 64

We can do the same thing for the 2nd row and 1st column:

(4, 5, 6) • (7, 9, 11) = 4×7 + 5×9 + 6×11 
    = 139

And for the 2nd row and 2nd column:

(4, 5, 6) • (8, 10, 12) = 4×8 + 5×10 + 6×12 
    = 154

And we get:



DONE!