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

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matrices to solve systems of two
simultaneous equations. Inverse

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matrices are always useful in
solving simultaneous equations,

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

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

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much on the previous video which
describe finding the determinant

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

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familiar with the ideas in that

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

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about cofactors and

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

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calculating determinants and we
saw in that video how every

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element in the matrix A has its

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

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

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

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

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

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

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

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

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

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

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not just this matrix is the
transpose of the cofactor

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matrix. So here's the cofactor

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matrix. To find the adjoint
matrix we have to transpose

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

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

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

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And the third row becomes
the third column.

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

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

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the determinant of a?

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Times by the adjoint matrix.

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I'm writing that using
this notation. A inverse

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is the value one
over determinant of a

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times by the adjoint
matrix of a.

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And that's the key result

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

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to find the inverse of our

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

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

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

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

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

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

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

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

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

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11, seven 9 -- 3421.

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What you should do is you
should check by doing matrix

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multiplication that when you
multiply original matrix A by

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

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multiplied by inverse that you
do indeed get the three by

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three identity matrix.

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

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inverse matrix to solve a set of

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simultaneous linear equations.
Here we have a set of three

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

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

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

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

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

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

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

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

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

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

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

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

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

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this matrix disks.

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Then the solution of this
equation ax equals B.

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

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

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

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

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matrix.

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Minus 2.

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8 -- 5 three
minus 11 seven.

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9 --

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

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a few minutes ago.

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

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20 one 5 --

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

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

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That's our first entry.

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

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7 * -- 1 is minus 7.

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

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

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

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We have 63 -- 6 D 2 which

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

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

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Which is XY
zed?

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

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

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= 3 Y equals one, zed equals

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

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

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

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

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

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

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well.

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So inverse matrices are really
important when it comes to

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solving simultaneous equations.
Now you'll notice that our

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formula for finding the inverse
for three by three matrix had

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the value 1 divided by the
determinant of a.

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

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

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

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

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have an inverse matrix and
inverse matrix just doesn't

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

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trying to workout inverse
matrices, the thing we should do

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first of all is workout the
determinant and check that it

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isn't going to turn out to be 0.