https:/.../matrixinversionof3x3matrixf61mb-aspect.mp4

We've already seen how to use
the inverse of two by two

matrices to solve systems of two
simultaneous equations. Inverse

matrices are always useful in
solving simultaneous equations,

and so we want to look at in
this video is how to find the

inverse of a three by three
matrix. This video builds very

much on the previous video which
describe finding the determinant

of a three by three matrix, and
it's important that you're

familiar with the ideas in that

video. Before you watch this one
in particular, you need to know

about cofactors and

determinants. Here's the Matrix
a that we saw in the video on

calculating determinants and we
saw in that video how every

element in the matrix A has its

own cofactor. And the Co
factor is just a value, a

single number, and what I've
done here is I've assembled

all those cofactors into a
matrix that we've called see.

So for instance, the cofactor
of elements 7 is minus two,

and the cofactor Element 4 is
7, and so it goes on.

Now what we want to find the
inverse of matrix A. We have to

use this matrix C, but not quite
how it is at the moment. What we

have to do is we use it to
create something called the

adjoint matrix and we call the
adjoint matrix adj adj for a

joint. The adjoint of matrix A.
This is true for all matrices,

not just this matrix is the
transpose of the cofactor

matrix. So here's the cofactor

matrix. To find the adjoint
matrix we have to transpose

this. That means we have to
change the rows into columns.

The columns into rows. So the
first row minus 239 becomes the

first column minus 239. The 2nd
row becomes the second column.

And the third row becomes
the third column.

So we found now the adjoint
matrix, then the formula for the

inverse matrix. Is the inverse
of matrix A? Is one over

the determinant of a?

Times by the adjoint matrix.

I'm writing that using
this notation. A inverse

is the value one
over determinant of a

times by the adjoint
matrix of a.

And that's the key result

in finding. The inverse of any
matrix and will use that result

to find the inverse of our

matrix A. Here's our Matrix A
and we've worked out the adjoint

matrix. The formula for the
inverse is the inverse of A is

one over the determinant value
times by the adjoint of a. Now

in the video where we worked out
the determinant of this matrix,

we found the determinant of this
matrix A is equal to 1.

So in this case, the value 1
divided by the determinant of a

is just 1 / 1 which is 1. So
in this case a inverse is 1

times the joint of a or just the
adjoint of a. So a inverse turns

out to be just this matrix here
minus two 8 -- 5, three minus

11, seven 9 -- 3421.

What you should do is you
should check by doing matrix

multiplication that when you
multiply original matrix A by

this matrix we've just found
here a inverse, so we do a

multiplied by inverse that you
do indeed get the three by

three identity matrix.

What I want to go on to do now
is to show how we can use this

inverse matrix to solve a set of

simultaneous linear equations.
Here we have a set of three

simultaneous equations, 7X plus
two Y + Z = 21, three y --

8 = 5 minus three X + 4 Y
minus two, Z = -- 1. We want to

solve these define the values of
XY&Z. There are unknowns.

And we've seen how we can do. We
can represent these equations

using some key matrices, so we
have matrix A, which is the

matrix of coefficients 721
nought. 3 -- 1 -- 3, four and

minus 2. There's the question
here is not because we

haven't got any access, so
we've got like no XSS.

Then we have a matrix which is
just a single column, which is

the unknowns XY&Z and then I
have a separate matrix B which

again is just a single column
with the values from the right

hand side of the equations. So
in matrix form these equations

can be written as this matrix a
times this vector X is equal to

this matrix disks.

Then the solution of this
equation ax equals B.

We can find by multiplying
both sides by the inverse of A

to get that X = a inverse
times D. So to find our

solution XY and Z, we need to
find a inverse. And of course

we've just done that. We've
seen that a inverse is this

matrix.

Minus 2.

8 -- 5 three
minus 11 seven.

9 --

3421. So that's the matrix, a
inverse that we've just seen

a few minutes ago.

And so we have to do a inverse
times be. So here's be.

20 one 5 --

1. And so we do the matrix
multiplication 2 * 21 is minus

40, two 8 * 5 is 40 --
5 * -- 1 is +5.

That's our first entry.

I'm going along the 2nd row 3 *
2163 -- 11 * 5 is minus 50 five

7 * -- 1 is minus 7.

And along the last row, 9 *
21 is 189 -- 34 * 5

is minus 170 and 21 * --
1 is minus 21.

And so if we do the arithmetic,
we have 45 -- 42, which is 3.

We have 63 -- 6 D 2 which

is 1. And we have 189
-- 191, which is minus 2. So

the unknowns that we're trying
to find the column matrix X.

Which is XY
zed?

Is this column matrix three 1 --
2 so X is equal to 3, Y

is equal to 1 that is equal to
minus two. That's our solution X

= 3 Y equals one, zed equals

minus 2. And you can check if
you substitute these values

back into any of these
equations, you'll see that the

two sides do balance. So for
instance, just the second

equation with. Why is worn and
said he's minus two, we get 3

* 1 is 3 -- -- 2 three plus
two, which is indeed five you

submit those into the two.
You'll see that they work as

well.

So inverse matrices are really
important when it comes to

solving simultaneous equations.
Now you'll notice that our

formula for finding the inverse
for three by three matrix had

the value 1 divided by the
determinant of a.

Now that means that if the
determined today is zero, we

can't actually work that value
out because we can't divide by

zero. Now, when the determined
to the matrix is zero, we say

the matrix is singular and when
a matrix is singular, it doesn't

have an inverse matrix and
inverse matrix just doesn't

exist, and so we can't apply the
formula. So whenever we're

trying to workout inverse
matrices, the thing we should do

first of all is workout the
determinant and check that it

isn't going to turn out to be 0.