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One of the most important
applications of matrices is to

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finding the solution of a pair
of simultaneous equations. So in

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this video I'm going to show you
how we can use matrices to do

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just that. Let's look at this
pair of simultaneous equations

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here. X + 2 is four and three X
-- 5 Y is 1.

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Provided you understand how
matrices are multiplied

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together, you will be able to
write these equations in what's

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called matrix form. I'll show

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you how. Supposing we write down
the Matrix 1, two, 3 -- 5, and

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we multiply it by another matrix
XY. Let's just see how we can do

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that. So we have a two by two
matrix here and I formed it from

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the coefficients of X&Y in the
simultaneous equations 1, two, 3

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-- 5. I've written this matrix
here X&Y, which was which,

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contains the unknowns in the

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simultaneous equations. Now,
provided you can understand how

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to do matrix multiplication, you
realize that these can be

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multiplied together as follows.
You pair the Elements 1, two

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with XY and multiply the paired
elements together and then add.

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So it's 1 * X.

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Which is X added to 2 *
y which is 2 Y.

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And you'll notice that X + 2 Y
is the same as the quantity on

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the left hand side of the first

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equation. If we look at this
second row here and multiply it

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by the X and the Y 3 * X
-- 5 * y.

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And you'll see what we have

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here. Is the left hand side of
the second equation in the

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simultaneous equations? And then
we can write. This lot is equal

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to 4, one which is a matrix
formed by the right hand side

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numbers in the simultaneous

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equations. So you've really got
the simultaneous equations

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written down. In here you got X
+ 2 Y's four and three X -- 5 Y

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is 1. So these are the original
simultaneous equations, but in

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matrix form. If we call this

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matrix A. And this matrix
of unknowns. Let's call

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that capital X.

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And this matrix on the right
hand side. Let's call that B.

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This form now have here is
what we call the matrix form

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of the simultaneous equations.

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Let's think about which bits in
this in this in this matrix

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form. We know which bits we
don't know. Well, the matrix A1,

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two 3 -- 5 is known. It's just
the matrix of coefficients of

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X&Y in the original equations,
the matrix be on the right is

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this 141, which is just the
matrix formed by the right hand

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side of the simultaneous

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equations. This quantity,
capital X. This matrix here,

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which stands for little X
little. Y that's the unknown.

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That's the thing we're trying
to find, so this is the

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matrix of unknowns.

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So what we'd like to do is,
like, we'd like to rearrange

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this and get X equal to
something, because then we've

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got the solution of the
simultaneous equations. We've

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got the unknown on its on its
own on the left hand side. Now

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we can find this unknown matrix
X the matrix of unknowns using

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the inverse matrix. Let's see
how we can do that.

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So this is the matrix equation
we want to solve. Now you might

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be tempted to say if you want to
find X, then X is B / A, but

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remember there's no such thing
as matrix division. You can't

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divide matrix be by a matrix A
and instead we make use of the

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inverse matrix. What we'll do is
we'll take this matrix form and

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we multiply both sides of it by
the inverse of matrix A. So I'm

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going to take the left hand side
and multiply it by the inverse.

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Of a. And do the same to the
right hand side.

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So both sides of this equation
have been multiplied by 8 to the

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minus one. Now remember a very
important property of inverse

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matrices is that when you take A
to the minus one and multiply it

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by a, the result here.

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Is an identity matrix, so ETA
minus one A is an identity

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matrix, so even identity matrix
times X is A to the minus 1B.

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And further property that you'll
need to remember, which is very

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important, is that if you take
any matrix X and you multiply it

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by an identity matrix, it leaves
it identical to what it was

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before, so the left hand side
just simplifies this simply to

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X. So we've come to the
conclusion that if a X is B,

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then the unknown X is the
inverse of a * B. So this is the

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important result that will need

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to use. To solve a pair of
simultaneous equations, what

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will need to do is find the
inverse of the matrix of

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coefficients and multiply that
by the matrix formed by the

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elements by the numbers on the
right hand side of the

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simultaneous equations. So this
is the formula will need will do

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that in an example now.

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So let's continue with this
example. Here are the

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simultaneous equations that we
started with, and we've written

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them in matrix form like this.
That's in the form ax is B.

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Well, this is the matrix. A

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matrix of coefficients. This is
the matrix Capital X, the matrix

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of unknowns and B is the matrix
formed by the elements on the

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right hand side. What we need to
do to solve this is we need to

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find the inverse of a. So let's
start by writing down the

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inverse of this matrix here. Now
remember that the formula for

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the inverse matrix is as
follows. 1 / a D minus BC.

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That's 1 * -- 5, which is minus
5. Subtract VC's 2346.

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We interchange the A and the D.
So minus five will go up there

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and one will go down there and
we change the sign of the other

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two elements. So this is the
inverse of the matrix A.

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Let's just tidy it up a little
bit, minus 5 -- 6 is minus 11,

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so we've got minus and 11th.

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Of the matrix minus 5 -- 2 -- 3
and one and I'm going to leave

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it in that form. I don't need to
worry about taking the factor of

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minus 1 / 11 inside.

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Right, what we want to do now is
find the matrix of unknowns X,

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and remember that X is given by
A to the minus 1 * B.

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Eight of the minus one is now
minus and 11th.

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Times minus 5 -- 2
-- 3 one multiplied by

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matrix B which was four

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one. And you'll see that all we
need to do now to finish this

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off is debit of matrix
multiplication. Let's leave the

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minus 11 outside.

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And over here we've got minus 5
* 4, which is minus 20 added to

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minus 2 * 1. That's added to
minus two, so you've got minus

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22 there. And here pairing up
this road with this column we've

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got minus 3 * 4 is minus 12
added to 1 * 1, which is minus

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12 model on which is minus 11.

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Finally, just to finish this
off, every element inside the

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matrix has to be multiplied by
minus in 11th or in other words

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divided by minus 11. And if we
do that, this first entry will

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be minus 22 / -- 11, which is 2.

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And the second element is minus
11 / -- 11, which is 1. So

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that's the solution. This is the
matrix X, which you'll remember

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is the same as the matrix XY.
This means that our little X is

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2 and our little wise one, and
that's the solution of the

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simultaneous equations. Those
are the solutions of the two

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equations that we started with.

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Rather than just leave it there
like that, what you ought to do

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to do is check that this
solution is in fact correct by

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substituting X is too. And why
is 1 into these equations just

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to check it's right. So for
example, if you protect this

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two in here, you'll get two and
two ones are two there, two and

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two is 4, so it satisfies the
first equation. You should

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check that it satisfies the
second equation as well.