7.3   MATRIX MULTIPLICATION

Often we will find a need to multiply one matrix by another. To see this in
action, let us look at another simple “mixing” example.

In a candy store, “scrunches”, “munches,” and “chews” are on sale.

Also on sale are “Jumbo” bags each containing 2 scrunches, 3 munches,
and 4 chews, and “Giant” bags containing 5 scrunches, 6 munches and only
one chew. If I purchase 7 Jumbo bags and 8 Giant bags, how many of each
sweet have I bought?

The bag contents can be expressed algebraically as

and

or in matrix form as

Note that matrices do not have to be square, as long as the terms to be multiplied
correspond in number.

Now my purchase of 7 Jumbo bags and 8 Giant bags can be written as

or in grander form as the product of a row vector with a column vector:

But I can substitute for the J,G vector to obtain

To get numerical counts of scrunches, munches, and chews we have to calculate
the product of a numerical row vector with a numerical matrix. As before,
we march across the row(s) of the one on the left, taking the scalar product
with the columns on the right.

The answer is what common sense would give.

From 7 Jumbo bags, with scrunches at 2 to a bag, we find 7 times 2
scrunches.

From 8 Giant bags, we find 8 times 5 more, giving a grand total of 54.

The final answer is

that is 54 scrunches, 69 munches, and 36 chews.

Now the shop is selling an Easter bundle of 3 Jumbo bags and a Giant bag,
and still has in stock Christmas bundles of 2 Jumbo bags and 4 Giant bags.
If I buy five Easter packs and one Christmas pack, how many scrunches,
munches, and chews will I have?

As an exercise, write down the matrices involved and multiply them out
by the rules that we have found. (Your answer should be 89 scrunches + 105
munches + 77 chews.)

The mathematician will still worry about the order in which the matrix
multiplication is carried out. We must not alter the order of the matrices, but
we can group the pairs for calculation in either of two ways.

The Christmas and Easter bags can first be opened to reveal a total of
Jumbo and Giant bags, then these can be expanded into individual sweets.
Alternatively, work out the total of each sweet for a Christmas bag and for
an Easter bag first. The result must be the same. (Check it.)

Mathematicians would say that “multiplication of matrices is
associative:”