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.