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In this video I'm going to[br]explain what is meant by a

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matrix and introduce the[br]notation that we use when we're

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working with matrices. So let's[br]start by looking at what we mean

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by a matrix and matrix is a[br]rectangular pattern of numbers.

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Let's have an example.

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I'm writing down a[br]pattern of numbers.

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4 -- 113 and 9 and you see they[br]form a rectangular pattern and

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when we write down a matrix, we[br]usually enclose the numbers with

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some round brackets like that.[br]So that's our first example of a

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matrix. Let's have a look at[br]another example which has a

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different size. So suppose we[br]have the numbers 12.

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304 and again, this is a[br]rectangular pattern of

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numbers. I'll put them in[br]round brackets like that, and

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that's another example of a[br]matrix. Let's have some more.

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71

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minus three to four and four.

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And a final example.

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A half 00.

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03000 and let's[br]say nought .7.

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So here we have 4 examples[br]of matrices.

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They all have different sizes,[br]so let's look a little bit more

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about how we refer to the size[br]of a matrix. This first matrix

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here has got two rows and two[br]columns and we describe it as a

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two by two matrix. We write it[br]as two by two like that. That's

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just the size. When we write[br]down the size of a matrix, we

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always give the number of rows[br]first and the columns second.

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So this has two rows and two[br]columns.

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What about this matrix?

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This is got one row.

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And 1234 columns.

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So this is a 1 by 4 matrix, one

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row. And four columns.

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This matrix has got 123 rows.

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And two columns.

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So it's a three by two[br]matrix and the final example

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has got three rows.

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Three columns, so this is a[br]three by three matrix, so

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remember that we always give[br]the number of rows first and

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column 2nd.

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The other bit of notation that[br]we'll need is that we often use

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a capital letter to denote a[br]matrix, so we might call this

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first matrix here A.

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We might call the second 1B.

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This one C.

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And this 1D.

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So there we are four examples[br]of matrices, all of different

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sizes and we now know how to[br]describe a matrix in terms of

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the number of rows and the[br]number of columns that it

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has. Now that we've seen for[br]examples of specific

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matrices, let's look at how[br]we can write down a general

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matrix. Let's suppose this[br]matrix has the symbol A and

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let's suppose it's got M rows[br]and N columns, so it's an M

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by N matrix.

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The number that's in the first[br]row, first column of Matrix

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Capital A will write using a[br]little A and some subscripts 11

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where the first number refers to[br]the row label and the 2nd to the

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column label. So this is first[br]row first column.

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The second number will be a 12,[br]which corresponds to the first

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row, second column, and so on.[br]The next one will be a 1 three.

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And so on. Now in this matrix,[br]because it's an M by N matrix,

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it's got N columns, so the last[br]number in this first row will be

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a 1 N corresponding to 1st row.

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And column.

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What about the number in here?[br]Well, it's going to be in the

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2nd row first column, so we'll[br]call it a 2 one. That's the 2nd

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row, first column.

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The one here will be second[br]row, second column.

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2nd row, third column, and so on[br]until we get to the last number

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in this row. Which will be a 2 N[br]which corresponds to 2nd row,

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NTH column and so on. We can[br]build up the matrix like this.

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We put all these numbers in as[br]we want to be 'cause this matrix

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has got M rose.

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The last row here will have a[br]number AM one corresponding to M

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throw first column.

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NTH Row, second column and so on[br]all the way along until the last

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number here, which is in the M[br]throw and the NTH column. So

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we'll call that a MN and that's[br]the format in which we can write

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down a general matrix.

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Each of these numbers in the

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matrix. We call an element of[br]the matrix, so A1 one is the

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element in the first row, first[br]column. In general, the element

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AIJ will be the number that's in[br]the I throw and the J column.

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Now some of the matrices that[br]will come across occur so

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frequently or have special[br]properties that we give them

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special names. Let's have a look[br]at some of those.

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Let's go back and look again[br]at the Matrix A. We saw a

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few minutes ago.

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This matrix has got two rows.

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And two columns. So it's a two

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by two matrix. And a matrix[br]that's got the same number of

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rows and columns like this one,[br]has we call for obvious reasons

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a square matrix? So this is[br]the first example of a

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square matrix.

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Now we've already seen another[br]square matrix because the

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matrix D that we saw a few[br]minutes ago, which was this

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one.

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He's also a square matrix. This[br]one's got three rows.

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And three columns. It's a[br]three by three matrix, and

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because it's got the same[br]number of rows and columns,

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that's also a square matrix.

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Another term I'd like to[br]introduce is what's called a

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diagonal matrix.

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If we look again at the matrix[br]D, we'll see that it has some

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rather special property. This[br]diagonal, which runs from the

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top left to the bottom right, is[br]called the leading diagonal.

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And if you look carefully,[br]you'll see that all the elements

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that are not on the leading

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diagonal are zeros. 0000000 A[br]matrix for which all the

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elements off the leading[br]diagonal are zero, is called

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a diagonal matrix.

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There's another special sort of[br]diagonal matrix I'll introduce

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now. Let's call this one, I.

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Suppose this is a two by two[br]matrix with ones on the leading

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diagonal and zeros everywhere

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else. So this is a square

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matrix. It's diagonal because[br]everything off the leading

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diagonal is 0.

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And it's rather special because[br]on the leading diagonal, all the

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elements are one. Now a matrix[br]which has this property is

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called an identity matrix.

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Or a unit matrix?

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And when we're working with[br]matrices, it's usual to reserve

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the letter I for an identity[br]matrix. Now suppose we have a

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bigger identity matrix. Here's

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another one. Suppose we have a[br]three by three identity matrix.

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And again, notice that[br]it's square.

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It's diagonal and there are ones[br]only on the leading diagonal, so

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this is also an identity matrix.[br]But because this is a three by

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three and this ones are two by[br]two and we might not want to mix

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them up and might call this one[br]I2, because this is a two by two

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matrix and I might call this one[br]I3. But in both cases these are

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identity matrices and we'll see[br]that identity matrices have a

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very important role to play when[br]we look at matrix multiplication

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in a forthcoming video.