https:/.../simultaneousequationsf61mb-aspect.mp4

One of the most important
applications of matrices is to

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

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

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

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

Provided you understand how
matrices are multiplied

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

called matrix form. I'll show

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

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

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

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

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

contains the unknowns in the

simultaneous equations. Now,
provided you can understand how

to do matrix multiplication, you
realize that these can be

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

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

So it's 1 * X.

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

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

the left hand side of the first

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

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

And you'll see what we have

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

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

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

numbers in the simultaneous

equations. So you've really got
the simultaneous equations

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

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

matrix form. If we call this

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

that capital X.

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

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

of the simultaneous equations.

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

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

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

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

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

side of the simultaneous

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

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

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

matrix of unknowns.

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

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

got the solution of the
simultaneous equations. We've

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

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

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

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

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

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

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

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

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

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

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

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

minus one. Now remember a very
important property of inverse

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

by a, the result here.

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

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

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

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

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

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

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

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

important result that will need

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

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

coefficients and multiply that
by the matrix formed by the

elements by the numbers on the
right hand side of the

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

that in an example now.

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

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

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

Well, this is the matrix. A

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

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

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

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

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

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

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

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

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

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

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

so we've got minus and 11th.

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

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

minus 1 / 11 inside.

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

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

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

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

matrix B which was four

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

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

minus 11 outside.

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

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

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

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

12 model on which is minus 11.

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

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

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

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

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

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

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

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

simultaneous equations. Those
are the solutions of the two

equations that we started with.

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

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

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

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

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

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

check that it satisfies the
second equation as well.