Just as with adding matrices,
the sizes of the matrices matter when we are multiplying. For matrix multiplication
to work, the columns of the second matrix have to have the same number
of entries as do the rows of the first matrix.

AB
=

If, using the above matrices,
B
had had only two rows, its columns would have been too short to multiply
against the rows of A.
Then "AB"
would not have existed; the product would have been "undefined".
Likewise, if B
had had, say, four rows, or alternatively if A
had had two or four columns, then AB
would not have existed, because A
and B
would not have been the right sizes.

In other words, for AB
to exist (that is, for the very process of matrix multiplication to be
able to function sensibly), A
must have the same number of columns as B
has rows; looking at the matrices, the rows of A
must be the same length as the columns of B.

You can use this fact to
check quickly whether a given multiplication is defined. Write the product
in terms of the matrix dimensions. In the case of the above problem, A
is 2×3
and B is 3×2,
so AB
is (2×3)(3×2).
The middle values match:

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...so the multiplication
is defined. By the way, you will recall that AB,
the product matrix, was 2×2.
You can also see this on the dimensions:

Using this, you can see that
BA
must be a different matrix from AB,
because:

The product BA
is defined (that is, we can do the multiplication), but the product, when
the matrices are multiplied in this order, will be 3×3,
not 2×2.
In particular, matrix multiplication is not "commutative";
you cannot switch the order of the factors and expect to end up with the
same result. (You should expect to see a "concept" question
relating to this fact on your next test.)

Given the following
matrices, find the product BA.

Matrix multiplication is
probably the first time that the Commutative
Property has ever
been an issue. Remember when they made a big deal, back in middle school
or earlier, about how "ab
= ba" or "5×6
= 6×5"? That "rule"
probably seemed fairly stupid at the time, because you already knew that
order didn't matter in multiplication. Introducing you to those rules
back then was probably kind of pointless, since order didn't matter
for anything you were multiplying then. Well, now the Law of Commutativity
does matter, because order does matter for matrix multiplication.
Always keep in mind that, for matrices, AB
almost certainly does not equal BA.