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

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matrix and introduce the
notation that we use when we're

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

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by a matrix and matrix is a
rectangular pattern of numbers.

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Let's have an example.

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I'm writing down a
pattern of numbers.

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

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

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

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

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different size. So suppose we
have the numbers 12.

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304 and again, this is a
rectangular pattern of

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

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that's another example of a
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
say nought .7.

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So here we have 4 examples
of matrices.

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

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

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

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

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

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always give the number of rows
first and the columns second.

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So this has two rows and two
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
matrix and the final example

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has got three rows.

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Three columns, so this is a
three by three matrix, so

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remember that we always give
the number of rows first and

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column 2nd.

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

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a capital letter to denote a
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
of matrices, all of different

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

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the number of rows and the
number of columns that it

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

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

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

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let's suppose it's got M rows
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
row, first column of Matrix

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Capital A will write using a
little A and some subscripts 11

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where the first number refers to
the row label and the 2nd to the

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column label. So this is first
row first column.

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

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

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

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it's got N columns, so the last
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?
Well, it's going to be in the

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2nd row first column, so we'll
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
row, second column.

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

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

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

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We put all these numbers in as
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
number AM one corresponding to M

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throw first column.

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

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

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we'll call that a MN and that's
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
the matrix, so A1 one is the

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element in the first row, first
column. In general, the element

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

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Now some of the matrices that
will come across occur so

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frequently or have special
properties that we give them

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

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Let's go back and look again
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
that's got the same number of

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

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a square matrix? So this is
the first example of a

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square matrix.

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Now we've already seen another
square matrix because the

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

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one.

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

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

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because it's got the same
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
introduce is what's called a

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diagonal matrix.

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

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rather special property. This
diagonal, which runs from the

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

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And if you look carefully,
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
matrix for which all the

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elements off the leading
diagonal are zero, is called

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a diagonal matrix.

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There's another special sort of
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
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
everything off the leading

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diagonal is 0.

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

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elements are one. Now a matrix
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
matrices, it's usual to reserve

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the letter I for an identity
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
three by three identity matrix.

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And again, notice that
it's square.

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

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

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

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

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

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identity matrices and we'll see
that identity matrices have a

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very important role to play when
we look at matrix multiplication

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in a forthcoming video.