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.