Once you know how to multiply 2
matrices together, a natural

question to ask is whether or
not one matrix can be divided by

another matrix, and the answer
to that question is no, there is

no such thing as matrix

division. Nevertheless, there's
another operation that we can

introduce which plays a very
similar role to division, and

it's called finding the inverse
of a matrix. So in this video

I'm going to explain what we
mean by the inverse of a matrix.

To start with, let's just look
at these two matrices that I've

written down here, and let's
find the product of them. We

multiply these two together.

4 * 1 is 4 added to 3 *
-- 1 is 4 -- 3 which is 1.

4 * -- 3 is minus 12, three 4 +
12 -- 12 + 12 is 0.

1 * 1 is 1 and 1 * -- 1 is minus
one 1 -- 1 is or is not.

1 * -- 3 is minus three 1 * 4 is
four 4 -- 3 is 1.

The point I'm trying to make is
that when we multiply these two

matrices together, the answer
that we get is an identity

matrix. That point is important.

Suppose we do the
multiplication in the opposite

order, so 1 -- 3 -- 1 four
multiplied by 4311. So I've

changed the order of the
multiplication there.

One's voice 4 -- 3 * 1 is minus
three. 4 -- 3 is 1.

Once three is 3 -- 3 * 1 is
minus three, 3 -- 3 is nothing.

Minus 1 * 4 is minus 44144 and
the result of adding them

together is 0.

Minus 1 * 3 is minus 3414, four
4 -- 3 is 1, so again the result

of multiplying these two
together is an identity matrix.

This leads us to the following
definition for the inverse of a

matrix. If you've got one matrix
and you multiply it by another

matrix, and the answer is an
identity matrix, then the second

matrix is the inverse of the
first matrix and vice versa. The

first matrix is the inverse of
the second matrix. Now suppose

we have a matrix A and we
multiply it by its inverse and

we'll denote the inverse by A to
the power minus one. Now that's

not really a power. That does
not mean 1 / A.

This is the notation that
will use for the inverse of

the matrix A, so if you have
a matrix a multiplied by its

inverse.

The answer is an identity
matrix. That's what we mean

by an inverse matrix. The
same works the other way

around as well, so if you
started with an inverse

matrix and multiplied it by
the original matrix, the

answer two would be an
identity matrix, and that's

the important definition of
an identity matrix.

If we want to find an inverse of
the matrix, there's a simple

formula that exists when the
matrix is a two by two matrix.

So let's have a look at the

formula Now. Suppose we've
got the matrix ABCD.

The formula for the inverse
is as follows. The inverse,

which as I say is denoted by
A to the minus one.

Is given by 1 divided by.

The number is 8 * D. Subtract B
* C rather strange formula, but

you'll see how it develops in a
minute. 1 / a D minus BC

multiplied by the matrix that
you get when you interchange the

A and the D round. So the DB
comes up here and the A goes

down there. And we changed the
sign but leave leave these

elements in place. The BNC in
place, but change their sign to

minus BN minus C, and this
formula that we have here is the

formula. Then for the inverse of
the matrix A.

I'm going to apply this
formula to try and find the

inverse of a matrix that I'll
give you here. Now, supposing

the Matrix is a is 3142.

Now applying the formula, we'll
get that the inverse matrix is

equal to 1 / 8 * D. That's 3 *
2, which is 6 -- B * C, which is

1 * 4.

Multiplied by a matrix which you
get when you interchange the A

and the D. So the A and the D

swap round. So swapping the
three in the two round will

get two and three.

And changing the sign of the
other two elements, but leaving

them in the same place.

So this is the inverse of matrix

A. Let's just tidy it
up a little bit 1 / 6

-- 4 is 1 / 2 or 1/2.

And we could leave the answer
like that. Or we can multiply

the factor of 1/2 inside,
multiplying every element inside

by the half. But I'm going to
leave it like that for now.

So this is the inverse of matrix
A. If we want to check whether

it's whether it's the right
answer or not, all we have to do

is multiply these two together
and the answer that we should

get should be an identity
matrix. So let's just check that

what I'll do is I'll workout A
to the minus 1 * A.

So here's A to the minus
one we've just found.

And I'm going to multiply it by
a, which was three 142.

Let's leave the half
outside for now.

To do the matrix multiplication,
here we've got 2, three or six.

Added 2 -- 1 * 4 that's 6 -- 4
which is 2.

2 ones or two.

Add it to minus 1 * 2. That's 2
-- 2, which is nothing.

Minus 4 * 3.

Plus 3 * 4, which is minus 12 +
12, which is nothing and finally

minus 4 * 1, which is minus four
added to 3, two six 6 -- 4 is 2.

And finally, if we multiply each
element by the factor of 1/2

outside, you'll see that we get
the identity matrix, so that's a

verification that this matrix we
have found here is the inverse

of the matrix we started with.

For yourself, you could verify
that we get the same results by

multiplying them the opposite
way around. Finding a time Zeta

minus one, and you'll still find
that you get an identity matrix,

and that's how you use the
formula to find the inverse of a

two by two matrix.

Now there are certain situations
where this won't work, and the

reason why it won't work is
obtained by looking at this

quantity in the formula here.

If it transpires that a D -- B C

is 0. Would be trying to divide
by zero here.

And division by zero is never

possible. So what this means is,
if ABC&D are such that a D -- B

C is 0.

The inverse of this matrix
will not exist.

Now you may remember that the
quantity AD minus BC we've met

before. It's called the
determinant of this matrix A.

So whenever the determinant of
the matrix A is 0.

The inverse won't exist.

Equivalently, whenever A is a
singular matrix, the inverse

won't exist. Let's just have a
quick look at an example of that

sort, supposing we had a matrix
A which was 326 fourth.

If we find a D minus BC for this
matrix, that's 3 * 4, which is

12. Subtract 2 * 6 which is 1212
-- 12 is zero. We see that the

determinant of a is 0.

This is a singular matrix, and
so because it's a singular

matrix, no inverse matrix will

exist. So the point to remember
is that not all two by two

matrices will have an inverse.
Those that don't have an inverse

accord singular matrices. But
when a matrix does have an

inverse, this formula will
enable you to find it.