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Once you know how to multiply 2
matrices together, a natural
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question to ask is whether or
not one matrix can be divided by
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another matrix, and the answer
to that question is no, there is
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no such thing as matrix
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division. Nevertheless, there's
another operation that we can
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introduce which plays a very
similar role to division, and
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it's called finding the inverse
of a matrix. So in this video
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I'm going to explain what we
mean by the inverse of a matrix.
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To start with, let's just look
at these two matrices that I've
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written down here, and let's
find the product of them. We
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multiply these two together.
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4 * 1 is 4 added to 3 *
-- 1 is 4 -- 3 which is 1.
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4 * -- 3 is minus 12, three 4 +
12 -- 12 + 12 is 0.
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1 * 1 is 1 and 1 * -- 1 is minus
one 1 -- 1 is or is not.
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1 * -- 3 is minus three 1 * 4 is
four 4 -- 3 is 1.
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The point I'm trying to make is
that when we multiply these two
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matrices together, the answer
that we get is an identity
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matrix. That point is important.
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Suppose we do the
multiplication in the opposite
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order, so 1 -- 3 -- 1 four
multiplied by 4311. So I've
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changed the order of the
multiplication there.
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One's voice 4 -- 3 * 1 is minus
three. 4 -- 3 is 1.
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Once three is 3 -- 3 * 1 is
minus three, 3 -- 3 is nothing.
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Minus 1 * 4 is minus 44144 and
the result of adding them
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together is 0.
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Minus 1 * 3 is minus 3414, four
4 -- 3 is 1, so again the result
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of multiplying these two
together is an identity matrix.
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This leads us to the following
definition for the inverse of a
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matrix. If you've got one matrix
and you multiply it by another
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matrix, and the answer is an
identity matrix, then the second
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matrix is the inverse of the
first matrix and vice versa. The
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first matrix is the inverse of
the second matrix. Now suppose
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we have a matrix A and we
multiply it by its inverse and
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we'll denote the inverse by A to
the power minus one. Now that's
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not really a power. That does
not mean 1 / A.
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This is the notation that
will use for the inverse of
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the matrix A, so if you have
a matrix a multiplied by its
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inverse.
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The answer is an identity
matrix. That's what we mean
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by an inverse matrix. The
same works the other way
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around as well, so if you
started with an inverse
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matrix and multiplied it by
the original matrix, the
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answer two would be an
identity matrix, and that's
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the important definition of
an identity matrix.
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If we want to find an inverse of
the matrix, there's a simple
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formula that exists when the
matrix is a two by two matrix.
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So let's have a look at the
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formula Now. Suppose we've
got the matrix ABCD.
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The formula for the inverse
is as follows. The inverse,
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which as I say is denoted by
A to the minus one.
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Is given by 1 divided by.
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The number is 8 * D. Subtract B
* C rather strange formula, but
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you'll see how it develops in a
minute. 1 / a D minus BC
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multiplied by the matrix that
you get when you interchange the
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A and the D round. So the DB
comes up here and the A goes
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down there. And we changed the
sign but leave leave these
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elements in place. The BNC in
place, but change their sign to
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minus BN minus C, and this
formula that we have here is the
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formula. Then for the inverse of
the matrix A.
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I'm going to apply this
formula to try and find the
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inverse of a matrix that I'll
give you here. Now, supposing
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the Matrix is a is 3142.
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Now applying the formula, we'll
get that the inverse matrix is
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equal to 1 / 8 * D. That's 3 *
2, which is 6 -- B * C, which is
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1 * 4.
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Multiplied by a matrix which you
get when you interchange the A
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and the D. So the A and the D
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swap round. So swapping the
three in the two round will
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get two and three.
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And changing the sign of the
other two elements, but leaving
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them in the same place.
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So this is the inverse of matrix
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A. Let's just tidy it
up a little bit 1 / 6
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-- 4 is 1 / 2 or 1/2.
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And we could leave the answer
like that. Or we can multiply
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the factor of 1/2 inside,
multiplying every element inside
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by the half. But I'm going to
leave it like that for now.
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So this is the inverse of matrix
A. If we want to check whether
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it's whether it's the right
answer or not, all we have to do
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is multiply these two together
and the answer that we should
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get should be an identity
matrix. So let's just check that
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what I'll do is I'll workout A
to the minus 1 * A.
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So here's A to the minus
one we've just found.
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And I'm going to multiply it by
a, which was three 142.
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Let's leave the half
outside for now.
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To do the matrix multiplication,
here we've got 2, three or six.
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Added 2 -- 1 * 4 that's 6 -- 4
which is 2.
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2 ones or two.
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Add it to minus 1 * 2. That's 2
-- 2, which is nothing.
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Minus 4 * 3.
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Plus 3 * 4, which is minus 12 +
12, which is nothing and finally
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minus 4 * 1, which is minus four
added to 3, two six 6 -- 4 is 2.
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And finally, if we multiply each
element by the factor of 1/2
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outside, you'll see that we get
the identity matrix, so that's a
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verification that this matrix we
have found here is the inverse
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of the matrix we started with.
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For yourself, you could verify
that we get the same results by
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multiplying them the opposite
way around. Finding a time Zeta
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minus one, and you'll still find
that you get an identity matrix,
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and that's how you use the
formula to find the inverse of a
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two by two matrix.
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Now there are certain situations
where this won't work, and the
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reason why it won't work is
obtained by looking at this
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quantity in the formula here.
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If it transpires that a D -- B C
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is 0. Would be trying to divide
by zero here.
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And division by zero is never
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possible. So what this means is,
if ABC&D are such that a D -- B
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C is 0.
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The inverse of this matrix
will not exist.
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Now you may remember that the
quantity AD minus BC we've met
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before. It's called the
determinant of this matrix A.
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So whenever the determinant of
the matrix A is 0.
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The inverse won't exist.
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Equivalently, whenever A is a
singular matrix, the inverse
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won't exist. Let's just have a
quick look at an example of that
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sort, supposing we had a matrix
A which was 326 fourth.
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If we find a D minus BC for this
matrix, that's 3 * 4, which is
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12. Subtract 2 * 6 which is 1212
-- 12 is zero. We see that the
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determinant of a is 0.
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This is a singular matrix, and
so because it's a singular
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matrix, no inverse matrix will
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exist. So the point to remember
is that not all two by two
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matrices will have an inverse.
Those that don't have an inverse
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accord singular matrices. But
when a matrix does have an
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inverse, this formula will
enable you to find it.