In this second of the videos on
matrix multiplication, we're
going to delve a little bit more
deeply into matrix
multiplication and look at some
of the properties and the
conditions under which different
sorts of multiplication can be
carried out. Let's start by
looking at looking at a specific
example in this example. Here
I've written down to matrices
M&N. And let's look at the sizes
of these matrices. The first
matrix M. Is a three row
three column matrix, so
it's three by three.
And the second matrix N is 3
rows, two columns.
So it's a 3 by 2.
And we notice that these
numbers are the same.
The number of columns in the
first is the same as the number
of rows in the second, so we can
perform this matrix
multiplication and the size of
the answer will be a three by
two matrix. So right at the
start we know the size of the
answer. It's going to have three
rows and two columns just like
this one had. So the shape
of the answer is.
Like we have here and we're
looking for these 6 numbers.
In the product.
Let's try and work it out.
To find the number that's in the
first row, first column. We work
with the first row of the first
matrix and the first column of
the Second Matrix.
What we want is 3
* 1 which is 3.
2 * -- 2 which is minus 4 and 1
* 3, which is 3. So we've got 3
ones or three.
2 * -- 2 is minus 4 and 1
* 3 is 3. We multiply the
paired elements together and
add the result.
When we come to the first row,
second column, we work with the
first row here and the second
column here. And again, pairing
off 3 * -- 2 is minus 6.
2 * 3.
Is 6.
1 * -- 4 is minus 4.
So in each case, we're
multiplying the paired elements
together and adding the results.
When we want the element that's
going in here, which is in the
2nd row first column of the
answer, we work with the 2nd
row, first column of the given
matrices 4 * 1 is 4.
Minus 3 * -- 2 is +6.
2 * 3 is 6.
And continuing in the same way,
the answer that goes in the 2nd
row, second column comes from
taking the 2nd row, second
column. 4 * -- 2 is minus 8.
Minus 3 * + 3 is minus 9.
2 * -- 4 is minus 8.
And finally on the last row
to find the element in the
first row. Sorry the 3rd row
first column will work with
the 3rd row, First Column, 5
ones of five.
4 * -- 2 is minus 8.
3 * 3 is 9.
And similarly to find the last
element, it will be 5 * -- 2,
which is minus 10.
4 * 3 is 12 and 3 * --
4 is minus 12.
And if we just tidy up what
we've got, we'll have 336
subtract 4, which is 2.
Minus 6 + 6 zero subtract 4 is
minus 4. Four and
six is 10 and 616.
Minus 8 -- 9 --
8 is minus 25.
5 subtract 8 + 9.
6th
and minus 10 + 12 -- 12 is minus
10. And this is the result
of multiplying these two
matrices together.
What about if we try and
multiply the two
matrices together the
opposite way round?
Suppose we try and
workout N * M.
Now, in this case the size of
the first matrix here is 3 rows
and two columns, so that's a
three by two and the size of the
second matrix is 3 by 3, three
rows, three columns.
And what we observe now is
that these two numbers here
are not the same, they are not
equal. That means that we
cannot do the matrix
multiplication in the order
that I've written it down
here. That matrix product
doesn't exist. So this is the
first point. I'd like to make
that even when you can find a
matrix product by multiplying
two matrices together, it
matters very much. The order
in which you write them down.
It may be possible to workout
a product one way, but not
another way. Let's look at
some more examples.
Suppose we've got two matrices
C&D as I've written them down
here, I'm going to try to work
out the product C * D.
And I'll also try and workout
the product D times. See if
either of these exist.
But in the first case, we've got
a two row three column matrix.
And in the second example
here, within the Second matrix
here we've got three rows into
two columns, so we can in fact
work this product out because
these numbers are the same and
the result will be a two by
two matrix. So the shape of
the answer will be 2 rows and
two columns.
If we try and do this the other
way round, D * C, The first
matrix Now has got three rows
and two columns. It's a three by
two matrix and the second one's
got two rows and three columns.
It's a two by three matrix.
So you can. You can see that we
can still work it out because
these two numbers are still the
same 2 into the same, but this
time the result is going to be a
three by three matrix, so it's
going to be a bigger matrix with
three rows and three columns.
We can use the process that we
evaluate that we worked on
before to evaluate the elements
in the these matrices. So for
example, the element that goes
in here is 1 * 3 + 2 * 5 added
to 3 * -- 1, which is 10.
And you can check for yourself
that the remaining elements are
131 and minus 11.
So it's possible to workout
C * D and the answer is a
two by two matrix.
When we do it the other way
round, let's take an element
here. Let's take the elements in
the first row, first column and
we obtain the answer by working
with the first row, first
column. Here, that's three
times, one is 3 added to minus 7
* 4. That's three added to minus
28, which is minus 25.
And you can proceed in the same
way to fill out this resulting
matrix and the numbers. You'll
get a -- 25 -- 29 -- 33.
9. 1521
789
The important point that I want
to make here is that when you
multiply C * D together.
It may be possible to also find
D * C, But the answers that you
get may have completely
different sizes. It's certainly
not true that CD is the same as
DC, so one of the observations
we take away straight away is
that in general CD is not equal
to DC. Even in situations where
both of these products do exist,
we say that matrix
multiplication is not
commutative. In general, it
really doesn't matter the order
in which you carry out the
multiplication. Now that we know
how to multiply 2 matrices
together, I'm going to show you
an important property of
identity matrices. Suppose we
have a two by two identity
matrix, that's 1001.
And suppose we have a second
matrix, two 3 -- 4 and seven.
And suppose I want to multiply
these two together.
The identity matrix is
certainly a two by two matrix,
and this matrix is also a two
by two matrix. So because
these numbers are the same, we
can actually workout the
product and the answer is also
a two by two matrix. So the
answer has this sort of shape
with four elements in there.
To get the first element in the
answer, we want to pair 10 with
2 -- 4, multiply the paired
elements together and add so we
get 1 * 2 is 2 added to 0 * --
4. Which is just 1 * 2 is 2.
To get this element here, we
want 1 * 3 which is 3 added to 0
* 7, which is just three.
To get the element in here, we
want to pair 01 with two and
minus four, so it's 0 * 2, which
is nothing 1 * -- 4 is minus 4,
so we just get minus 4.
And finally, the last element is
0 times. Three, which is nothing
1 * 7 is 7, so that's our
answer. And if you look at the
answer you'll see the answer is
identical to the matrix we
started with here. In other
words, multiplying a matrix by
an identity matrix when this
multiplication is possible
leaves an answer which is
identical to the matrix you
started with, and that's a very
important property of identity
matrices. The same result occurs
if we do the multiplication the
other way round. If we take two
3 -- 4 seven and we multiply it
by the identity matrix, one
nought nought one will find.
It's also possible, and if you
go through the operation 2 * 1
is 2 three times. Nothing is
nothing. The result there is 2.
Two times nothing
is nothing 313.
Minus 4 * 1 added to 7 times
nought is minus 4.
And minus four times North,
which is nothing added to 717
and you'll see again this answer
here is the same as this matrix
here. So that's very important
property to remember when you
multiply a matrix by an identity
matrix, it leaves the original
matrix unaltered, identical to
what it was before.
The same works even if we
haven't got square
matrices. Suppose we have
this identity matrix.
And we multiply, for example by
the Matrix 78.
Well, this has got one row and
two columns. It's a one by two
matrix. This is got two rows,
two columns, so we can perform
the matrix multiplication and
the result is going to be a
one by two matrix that's the
same shape as the one we
started with.
And if we carry out the
operations, it's 7 * 1, which
is 7 added to 8 times
nothing, which is nothing. So
the result is just 7th.
7 times and nothing is nothing
and 8 * 1 is 8, so it's just
eight. And again, this answer 7
eight is the same as the matrix
we started with over here. So
that's just to reinforce the
message that multiplying by an
identity matrix leaves the
original matrix unaltered.