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.