We call the actual numbers that occur in a given matrix its
entries. So the last exchange can be stated: “What was the entry
in the second row and ﬁrst column? Oh, it was a 0?” Or when you
get good at it, you can even say “What was the (2,1)-entry?”
Because the entries in a matrix are elements of a number system,
they can be added, subtracted, and multiplied. This allows us to
deﬁne multiplication of matrices, and thus we will obtain plenty of
“standard” groups to use in our representations.
Here is the all-important deﬁnition of how to multiply two
matrices. First, we have to tell you that any two matrices cannot
necessarily be multiplied. They must both have entries from the
same number system. For instance, both could be Z-matrices, or
Q-matrices, or C-matrices.
The sizes of the matrices also have to match in a particular way.
Remember that an m-by-n matrix is a rectangular array of numbers
with m rows and n columns. Let A be an m-by-nR-matrix and
B a p-by-qR-matrix. Then we can form and compute the product
AB (in that order!) if and only if n = p. In words: The number
of columns of the ﬁrst matrix must be the same as the number of
rows of the second matrix. You will see why as soon as we explain
how to multiply.
A mnemonic for this rule is that the numbers
in the middle have to “cancel out.” In fact, an m-by-n matrix
times an n-by-q matrix will be an m-by-q matrix, the n’s having
It will be helpful ﬁrst to describe the dot product of a row and
column of the same sizes. We assume the number system is ﬁxed
for this discussion, so that we can add and multiply entries. The dot
product of a row of length n and a column of length n is obtained
by multiplying the pairs of entries in corresponding positions in
You may have noticed that we do not use any symbol for the group law of multiplying
matrices. We just juxtapose the symbols of the two matrices, as in AB. This notation is