0:00:05.140,0:00:08.210
In this second of the videos on[br]matrix multiplication, we're

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going to delve a little bit more[br]deeply into matrix

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multiplication and look at some[br]of the properties and the

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conditions under which different[br]sorts of multiplication can be

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carried out. Let's start by[br]looking at looking at a specific

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example in this example. Here[br]I've written down to matrices

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M&N. And let's look at the sizes[br]of these matrices. The first

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matrix M. Is a three row[br]three column matrix, so

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it's three by three.

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And the second matrix N is 3[br]rows, two columns.

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So it's a 3 by 2.

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And we notice that these[br]numbers are the same.

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The number of columns in the[br]first is the same as the number

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of rows in the second, so we can[br]perform this matrix

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multiplication and the size of[br]the answer will be a three by

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two matrix. So right at the[br]start we know the size of the

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answer. It's going to have three[br]rows and two columns just like

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this one had. So the shape[br]of the answer is.

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Like we have here and we're[br]looking for these 6 numbers.

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In the product.

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Let's try and work it out.

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To find the number that's in the[br]first row, first column. We work

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with the first row of the first[br]matrix and the first column of

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the Second Matrix.

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What we want is 3[br]* 1 which is 3.

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2 * -- 2 which is minus 4 and 1[br]* 3, which is 3. So we've got 3

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ones or three.

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2 * -- 2 is minus 4 and 1[br]* 3 is 3. We multiply the

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paired elements together and[br]add the result.

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When we come to the first row,[br]second column, we work with the

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first row here and the second

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column here. And again, pairing[br]off 3 * -- 2 is minus 6.

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2 * 3.

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Is 6.

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1 * -- 4 is minus 4.

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So in each case, we're[br]multiplying the paired elements

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together and adding the results.

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When we want the element that's[br]going in here, which is in the

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2nd row first column of the[br]answer, we work with the 2nd

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row, first column of the given[br]matrices 4 * 1 is 4.

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Minus 3 * -- 2 is +6.

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2 * 3 is 6.

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And continuing in the same way,[br]the answer that goes in the 2nd

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row, second column comes from[br]taking the 2nd row, second

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column. 4 * -- 2 is minus 8.

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Minus 3 * + 3 is minus 9.

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2 * -- 4 is minus 8.

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And finally on the last row[br]to find the element in the

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first row. Sorry the 3rd row[br]first column will work with

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the 3rd row, First Column, 5[br]ones of five.

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4 * -- 2 is minus 8.

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3 * 3 is 9.

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And similarly to find the last[br]element, it will be 5 * -- 2,

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which is minus 10.

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4 * 3 is 12 and 3 * --[br]4 is minus 12.

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And if we just tidy up what[br]we've got, we'll have 336

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subtract 4, which is 2.

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Minus 6 + 6 zero subtract 4 is

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minus 4. Four and[br]six is 10 and 616.

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Minus 8 -- 9 --[br]8 is minus 25.

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5 subtract 8 + 9.

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6th

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and minus 10 + 12 -- 12 is minus

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10. And this is the result[br]of multiplying these two

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matrices together.

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What about if we try and[br]multiply the two

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matrices together the[br]opposite way round?

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Suppose we try and[br]workout N * M.

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Now, in this case the size of[br]the first matrix here is 3 rows

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and two columns, so that's a[br]three by two and the size of the

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second matrix is 3 by 3, three[br]rows, three columns.

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And what we observe now is[br]that these two numbers here

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are not the same, they are not[br]equal. That means that we

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cannot do the matrix[br]multiplication in the order

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that I've written it down[br]here. That matrix product

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doesn't exist. So this is the[br]first point. I'd like to make

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that even when you can find a[br]matrix product by multiplying

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two matrices together, it[br]matters very much. The order

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in which you write them down.[br]It may be possible to workout

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a product one way, but not[br]another way. Let's look at

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some more examples.

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Suppose we've got two matrices[br]C&D as I've written them down

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here, I'm going to try to work[br]out the product C * D.

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And I'll also try and workout[br]the product D times. See if

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either of these exist.

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But in the first case, we've got[br]a two row three column matrix.

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And in the second example[br]here, within the Second matrix

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here we've got three rows into[br]two columns, so we can in fact

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work this product out because[br]these numbers are the same and

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the result will be a two by[br]two matrix. So the shape of

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the answer will be 2 rows and[br]two columns.

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If we try and do this the other[br]way round, D * C, The first

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matrix Now has got three rows[br]and two columns. It's a three by

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two matrix and the second one's[br]got two rows and three columns.

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It's a two by three matrix.

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So you can. You can see that we[br]can still work it out because

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these two numbers are still the[br]same 2 into the same, but this

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time the result is going to be a[br]three by three matrix, so it's

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going to be a bigger matrix with[br]three rows and three columns.

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We can use the process that we[br]evaluate that we worked on

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before to evaluate the elements[br]in the these matrices. So for

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example, the element that goes[br]in here is 1 * 3 + 2 * 5 added

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to 3 * -- 1, which is 10.

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And you can check for yourself[br]that the remaining elements are

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131 and minus 11.

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So it's possible to workout[br]C * D and the answer is a

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two by two matrix.

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When we do it the other way[br]round, let's take an element

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here. Let's take the elements in[br]the first row, first column and

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we obtain the answer by working[br]with the first row, first

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column. Here, that's three[br]times, one is 3 added to minus 7

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* 4. That's three added to minus[br]28, which is minus 25.

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And you can proceed in the same[br]way to fill out this resulting

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matrix and the numbers. You'll[br]get a -- 25 -- 29 -- 33.

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9. 1521[br]789

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The important point that I want[br]to make here is that when you

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multiply C * D together.

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It may be possible to also find[br]D * C, But the answers that you

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get may have completely[br]different sizes. It's certainly

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not true that CD is the same as[br]DC, so one of the observations

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we take away straight away is[br]that in general CD is not equal

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to DC. Even in situations where[br]both of these products do exist,

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we say that matrix[br]multiplication is not

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commutative. In general, it[br]really doesn't matter the order

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in which you carry out the

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multiplication. Now that we know[br]how to multiply 2 matrices

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together, I'm going to show you[br]an important property of

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identity matrices. Suppose we[br]have a two by two identity

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matrix, that's 1001.

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And suppose we have a second[br]matrix, two 3 -- 4 and seven.

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And suppose I want to multiply[br]these two together.

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The identity matrix is[br]certainly a two by two matrix,

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and this matrix is also a two[br]by two matrix. So because

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these numbers are the same, we[br]can actually workout the

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product and the answer is also[br]a two by two matrix. So the

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answer has this sort of shape[br]with four elements in there.

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To get the first element in the[br]answer, we want to pair 10 with

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2 -- 4, multiply the paired[br]elements together and add so we

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get 1 * 2 is 2 added to 0 * --

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4. Which is just 1 * 2 is 2.

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To get this element here, we[br]want 1 * 3 which is 3 added to 0

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* 7, which is just three.

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To get the element in here, we[br]want to pair 01 with two and

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minus four, so it's 0 * 2, which[br]is nothing 1 * -- 4 is minus 4,

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so we just get minus 4.

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And finally, the last element is

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0 times. Three, which is nothing[br]1 * 7 is 7, so that's our

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answer. And if you look at the[br]answer you'll see the answer is

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identical to the matrix we[br]started with here. In other

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words, multiplying a matrix by[br]an identity matrix when this

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multiplication is possible[br]leaves an answer which is

0:10:16.646,0:10:20.078
identical to the matrix you[br]started with, and that's a very

0:10:20.078,0:10:21.326
important property of identity

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matrices. The same result occurs[br]if we do the multiplication the

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other way round. If we take two[br]3 -- 4 seven and we multiply it

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by the identity matrix, one[br]nought nought one will find.

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It's also possible, and if you[br]go through the operation 2 * 1

0:10:38.404,0:10:42.340
is 2 three times. Nothing is[br]nothing. The result there is 2.

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Two times nothing[br]is nothing 313.

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Minus 4 * 1 added to 7 times[br]nought is minus 4.

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And minus four times North,[br]which is nothing added to 717

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and you'll see again this answer[br]here is the same as this matrix

0:11:00.694,0:11:04.204
here. So that's very important[br]property to remember when you

0:11:04.204,0:11:08.065
multiply a matrix by an identity[br]matrix, it leaves the original

0:11:08.065,0:11:10.873
matrix unaltered, identical to[br]what it was before.

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The same works even if we[br]haven't got square

0:11:14.135,0:11:16.200
matrices. Suppose we have[br]this identity matrix.

0:11:19.650,0:11:23.178
And we multiply, for example by[br]the Matrix 78.

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Well, this has got one row and[br]two columns. It's a one by two

0:11:28.702,0:11:32.038
matrix. This is got two rows,[br]two columns, so we can perform

0:11:32.038,0:11:35.096
the matrix multiplication and[br]the result is going to be a

0:11:35.096,0:11:38.432
one by two matrix that's the[br]same shape as the one we

0:11:38.432,0:11:38.988
started with.

0:11:40.360,0:11:43.708
And if we carry out the[br]operations, it's 7 * 1, which

0:11:43.708,0:11:46.777
is 7 added to 8 times[br]nothing, which is nothing. So

0:11:46.777,0:11:48.172
the result is just 7th.

0:11:49.200,0:11:54.015
7 times and nothing is nothing[br]and 8 * 1 is 8, so it's just

0:11:54.015,0:11:58.188
eight. And again, this answer 7[br]eight is the same as the matrix

0:11:58.188,0:12:01.719
we started with over here. So[br]that's just to reinforce the

0:12:01.719,0:12:04.608
message that multiplying by an[br]identity matrix leaves the

0:12:04.608,0:12:05.571
original matrix unaltered.