## https:/.../matrixmultiplicationf61mb-aspect.mp4

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One of the most important
operations we can do with
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matrices is to learn how to
multiply them together. That's
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what we're going to do now. And
when we multiply matrices
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together, we find that we
combine the elements in the two
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matrices in rather a strange
way, and the easiest way to
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explain that is by example, so
let's have a look.
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Suppose we've got a
row of a matrix 37.
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And we want to combine it or
multiply it with a column of
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another matrix 29.
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And what we do is we combine
these numbers in a rather
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strange way. What we do is we
pair off the elements in the row
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of the first matrix with the
column with the column in the
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Second matrix, and we pair them
off and we multiply the
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corresponding elements together.
So we pair off the three with
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the two. The seven with the 9.
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And we multiply the paired
elements together so we
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have 3 * 2.
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I mean multiply the
seven with the 9.
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And we add the results together.
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So we have 3 * 2 which is
6 and 7 * 9, which is 63
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and we get the answer 69.
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So we have this rather strange
way in which we have combined
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the elements in the row of the
first matrix with the column
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in the Second matrix.
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Let's have a look at another
example. Suppose we have a row
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which is 425.
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And we're going to learn how to
multiply it with a column 368.
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And again, what we do is we pair
the elements off elements in the
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row of the first matrix with the
column of the Second matrix. We
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have 4 * 3.
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2 * 6.
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5 * 8.
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together, so with 4 * 3 which is
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12 two times 6 which is 12 and 5
* 8 which is 40. And if we add
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these up will get 12 and 12 is
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24. And 40 which is 64.
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So this is a rather strange way
in which we've combined the
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elements in the first matrix
with the elements in the second
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matrix, but it's the basis of
matrix multiplication, as we'll
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see shortly Now, suppose we have
to general matrices A&B, say.
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And we want to find the
product of these two
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matrices. In other words, we
want to multiply A&B
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together.
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Now suppose that this matrix
A. The first matrix has P,
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rossion, Q columns, so it's
a P by Q matrix.
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And the second matrix be.
Let's suppose that Scott
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are rows and S columns,
so it's an arby S matrix.
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Now it turns out that we can
only form this product. We can
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only multiply the two matrices
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together. If the number of
columns in the first matrix,
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which is Q is the same as the
number of rows in the second
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matrix, these two numbers have
got to be the same. Q Must equal
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R and the reason for that will
become apparent when we start to
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do the calculation. But you've
got to be able to pair up the
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elements in the first matrix
with the elements in the Second
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matrix and will only be able to
do that if the number of columns
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in the first is the same as the
number of rows in the second.
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When that's the case, we can
actually find the product AB and
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And let's suppose this answer is
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matrix C and the size of matrix.
See, we can determine in advance
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from the sizes of matrix A&B,
the size of matrix C will be P
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by S. So it's got the same
number of rows as the first
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matrix and columns as the second
matrix, so this will be an R.
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This will be a P by S matrix.
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Let's have a look at a specific
example. Suppose we want to
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multiply the matrix 37.
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45
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by the Matrix 29.
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And the first question we should
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matrices have the right size so
that we can actually multiply
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them together? Well, this matrix
is a two row two column matrix.
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And the second matrix
is 2 rows one column.
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And we note that the number of
columns here in the first matrix
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is the same as the number of
rows in the second matrix. So
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these two numbers are the same,
so we can do this multiplication
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and the size of the answer. The
size of the result that will get
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is obtained by the number of
rows in the 1st and columns in
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the second. So the size of the
answer that we're looking for is
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a 2 by 1 matrix. So you see
right at the beginning we can
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tell how many. Elements are
going to be in our answer.
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There's going to be a number
there, and a number there so
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that we have a 2 by 1 matrix.
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Now to determine these
numbers, we use the same
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operations as we've just
seen. We take the first row
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here and we pair the elements
with those in the first
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column.
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We multiply the paired elements
together and add the result. So
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we want 3 * 2, which is 6.
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We want 7 * 9 which is 63 and
we add the results together. 6
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and 63 is 69, so the element
that goes in the first position
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in our answer is 69. That's 3 *
2 at 7 * 9.
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The element that's going in this
position here is obtained by
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working with the 2nd row and
this first column here.
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Again, we pair the elements up 4
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* 2. Which is 8th.
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5 * 9 which is 45, and
we add the results together. 45
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+ 8 is.
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53
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So the result is 6953, so the
result of multiplying these two
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matrices together is another
matrix which is a 2 by 1 matrix
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and the elements are obtained in
the way I've just shown you.
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Let's have a look at another
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example. This time I'm going to
try to multiply together the two
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matrices A&B where a is this two
by two Matrix 2453 and B. Is
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this two by two Matrix three 6
-- 1 nine?
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And again, the first question we
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these matrices have the right
size so that we can actually
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multiply them together? Well,
matrix a this matrix A is a two
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row two column matrix. So that's
two by two.
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Matrix B is 2 rows and two
columns, so that's also
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two by two.
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And you can see that these two
numbers are the same. That is,
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the number of columns in the
first is the same as the
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number of rows in the second.
So we can perform the matrix
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multiplication.
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The size of the answer we can
determine right at the start.
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The size of the matrix that we
get is determined by the number
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here and the number there two by
two. So what we can decide
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before we do any calculation at
all is that this answer matrix
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is a two by two matrix.
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Be 2 rows and two columns,
so we're looking for four
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numbers to pop in there.
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Let's try and figure out how we
work out, what the answer is.
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When we want to find the element
that goes in here, observe that
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this is the first row first
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And the number in the first
row first column comes from
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looking at the first row and
1st Column here.
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If we pair off the elements in
the first row and 1st Column
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will have 2 * 3 which is 6.
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4 * -- 1, which is minus four,
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When we come to this element
here, this element is in the
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first row, second column.
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So we use the first row, second
column in the original matrices.
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2 * 6 which is 12 and 4
* 9 Four nines of 36. And
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products together.
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When we want the element that's
in the 2nd row first column, we
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use the 2nd row in the first
matrix and 1st column in the
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Second Matrix. Again, pairing
the elements off 5 * 3 is 15.
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3 * -- 1 is minus three.
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elements together.
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Finally. The element that's in
the 2nd row, second column of
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the answer is obtained by using
the elements in the 2nd row of
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the first matrix, second column
of the Second matrix.
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5 * 6 which is 30 and 3 *
9, which is 27, and we add the
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paired elements together.
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So finally, just to tidy it
up, we've got 6 subtract 4
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which is 2.
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12 + 36, which is 48.
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15 subtract 3 which is 12 and 30
+ 27 which is 57, so we can find
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the matrix product AB in this
case and the result is a two by
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two matrix. Let's have a look
at another example. Suppose
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product of these two matrices.
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And again, we should ask all
these matrices of the
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appropriate size. The first
matrix here has two rows, one
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column, it's a 2 by 1 matrix.
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And the second matrix has two
rows and two columns, so it's a
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two by two matrix. Now in this
case, you'll see that the
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number of columns in the first
matrix is not the same as the
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number of rows in the second
matrix. Those two numbers are
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not equal, so we cannot
multiply these matrices
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together. We say the product of
these matrices doesn't exist,
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so we stop there. We can't
calculate that.
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Let's have another example.
Suppose we want to try to
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find the product of the
matrices 3214.
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With the matrix XY. Now this
is the first example we've
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symbols rather than numbers in
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our matrix, but the operation
the process is, the
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calculations are just the
same.
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First of all, we should ask can
we multiply these together? Are
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they of the right size?
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This is a two row two
column matrix.
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And this is a two row one
column matrix.
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And these numbers are the same
in here the number of columns in
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the first is the same as the
number of rows in the second. So
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we can actually perform the
matrix multiplication and the
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answer we get will be a 2 by 1
matrix, so we know the shape of
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the answer. It's a two row one
column matrix, so it looks the
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same shape as this one. This one
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here. Two rows, one column.
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Let's actually work out what the
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As before, we take the first row
and pair the elements with the
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first column. So it's 3
multiplied by X 2 * y and we add
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the resulting products, so we
get three X + 2 Y.
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The element that's down here,
which is in the 2nd row, first
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obtained by using the 2nd row in
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the first matrix and the first
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it's 1 * X which is X 4 * y,
which is 4 Y and we add the
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products together and we get X +
4 Y. So the result we found is a
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two row one column matrix.
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In this case, the answers got
symbols in as well, but that's
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the result of finding the
product of these two matrices.
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Now we can go on and look at
more examples and trying to
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find products of matrices of
different sizes and shapes,
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and we'll do some more of that
in the next video.
Title:
https:/.../matrixmultiplicationf61mb-aspect.mp4
Video Language:
English
Duration:
13:23
 mathcentre edited English subtitles for https:/.../matrixmultiplicationf61mb-aspect.mp4