https:/.../introductiontomatricesf61mb-aspect.mp4

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

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

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

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

Let's have an example.

I'm writing down a
pattern of numbers.

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

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

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

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

different size. So suppose we
have the numbers 12.

304 and again, this is a
rectangular pattern of

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

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

71

minus three to four and four.

And a final example.

A half 00.

03000 and let's
say nought .7.

So here we have 4 examples
of matrices.

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

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

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

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

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

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

So this has two rows and two
columns.

What about this matrix?

This is got one row.

And 1234 columns.

So this is a 1 by 4 matrix, one

row. And four columns.

This matrix has got 123 rows.

And two columns.

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

has got three rows.

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

remember that we always give
the number of rows first and

column 2nd.

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

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

first matrix here A.

We might call the second 1B.

This one C.

And this 1D.

So there we are four examples
of matrices, all of different

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

the number of rows and the
number of columns that it

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

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

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

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

by N matrix.

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

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

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

column label. So this is first
row first column.

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

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

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

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

a 1 N corresponding to 1st row.

And column.

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

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

row, first column.

The one here will be second
row, second column.

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

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

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

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

has got M rose.

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

throw first column.

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

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

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

down a general matrix.

Each of these numbers in the

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

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

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

Now some of the matrices that
will come across occur so

frequently or have special
properties that we give them

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

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

few minutes ago.

This matrix has got two rows.

And two columns. So it's a two

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

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

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

square matrix.

Now we've already seen another
square matrix because the

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

one.

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

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

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

that's also a square matrix.

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

diagonal matrix.

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

rather special property. This
diagonal, which runs from the

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

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

that are not on the leading

diagonal are zeros. 0000000 A
matrix for which all the

elements off the leading
diagonal are zero, is called

a diagonal matrix.

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

now. Let's call this one, I.

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

diagonal and zeros everywhere

else. So this is a square

matrix. It's diagonal because
everything off the leading

diagonal is 0.

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

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

called an identity matrix.

Or a unit matrix?

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

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

bigger identity matrix. Here's

another one. Suppose we have a
three by three identity matrix.

And again, notice that
it's square.

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

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

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

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

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

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

very important role to play when
we look at matrix multiplication

in a forthcoming video.