The formal definition of matrix-matrix multiplication:
Given an m × r matrix A and an r × n matrix B, the product matrix C = AB is an m
× n matrix
with entries ci,j such that
It is not immediately obvious what that definition is
telling you to do, although you can probably see right away
you’re multiplying elements of A and elements of B that correspond in some
However, having defined row-column multiplication, we can describe the process
of matrix-matrix multiplication in
the fol lowing way . What that definition is saying is: to get the i, jth element
of the product C = AB, multiply
the ith row of A with the jth column of B
For example, to get c1,1, multiply the first row of A with the first column of
To get c2,3, multiply the second row of A with the third row of B.
And so on ...
Sounds complicated? It’s tough to describe in symbols and in words , but makes
sense with an example. Go to the
next page and fill in the grid. There’s a screen in the clip that’ll do the
find the product C = AB.
Multiplying row 1 of A wit column 1 of B gives c1,1.
Multiplying row 1 of A with column 2 of B gives c1,2.
Multiplying row 1 of A with column 3 of B gives c1,3.
Multiplying row 2 of A with column 1 of B gives c2,1
Multiplying row 2 of A with column 2 of B gives c2,2
Multiplying row 2 of A with column 3 of B gives c2,3,
Multiplying row 3 of A with column 1 of B gives c3,1.
Multiplying row 3 of A with column 2 of B gives c3,2.
Multiplying row 3 of A with column 3 of B gives c3,3.
• Do A and B have to be the same size for multiplication to be defined?
• What does have to match up?
• There’s an easy way to remember this (and to also predict the size of the
resulting product matrix):