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.