CHAPTER 05.24: SYSTEM OF
EQUATIONS: Finding the inverse of a matrix by adjoints Example In
this segment we will take the inverse of a matrix by using the adjoints
method, let’s take an example here. Somebody asks you to find the inverse of
this matrix here: 25, 5, 1, 64, 8, 1, 144, 12, and 1. So find the inverse of
this matrix. That is the problem statement. So we know that in order to find
the A inverse-what we have to do is, we have to find the determinate of the A
matrix and then we also have to find the adjoint of
A. If we are able to do that, then we are able to find the inverse of the
matrix. So let’s look at a few steps. The determinate of A is given to you as
-84. We have done this in a previous segment. Let’s go and see how to find
the adjoint of A. As we said, adjoint
of A is going to be based on theco factors. Let’s
see, we are going to find cofactors C11. How do we find cofactor C11? Cofactor
C11 will be corresponding to the element A11, which is 25, right here. That
is defined as -1 raised I plus 1 times M11- well 1 plus 1 I should say-times
M11. M11 is the minor of the matrix corresponding to A11. So C11 is the first
row and first element of the cofactor matrix and that’s the equal to -1
raised to the power of I plus J. I plus J. M11 is the minor of the
–corresponding to the A11 element. So this gives us M11 itself. Now how do I
find M11? I find M11, the minor of the entry M11 by simply getting rid of the
first row and the first column. So get rid of the first row and first column.
So I get rid of the first column and the first row. First row and first
column. I am left with 8, 1, 12 and 1. And then I find the determinate of
whatever I am left with. I am left with the sub-matrix of 2 by 2. 8, 1, 12,
and 1. Finding the determinate of a 2 by 2 is pretty simple. Its 8 times 1
minus 12 times 1 and that gives you -4. So that is how I found out the C11,
which is equal to M11, which is equal to -4. This gives me C11is just -4.
Let’s go and find out another cofactor corresponding to, let’s suppose, C12
to A12. Let’s go ahead and find what C12 is. C12 would be -1 raised power I
plus J, which would be 1 plus 2. M12 and that would be –M12. Now
what is M12? M12 will be the minor of the matrix corresponding to the A12
element. But it will be the determinate of the matrix that is left over, once
I take the 1st row out and the 2nd column out. So if I take the 1st row out
and the 2nd column out, I am left with: 64, 1, 144, and 1. That is nothing
more than 64 times 1 minus 144 times 1. That is -80. So that is what M12 is,
so that means that C12 is equal to –M12. Since M12 is -80, C12 is going to be
80. So you get the point on how you are going to calculate the cofactor
matrix corresponding to each of the 9 elements, which you have in the A
matrix. We found C11 equal to -4. We found C12 equal to 80. We can similarly
find C13 to be -384 you can (4:22) C21 turns out to be 7. C22 turns out to be
-119. C23 turns out to be 420. Then C31 turns out to be -3, C32 is 39. C33
turns out to be-120. These are the cofactors which we get, corresponding to
the 9 elements of the A matrix. So now I can write my C matrix, which is my
cofactor matrix. So I just write it in the matrix form. -4, 80, -384. It’s
simply putting these elements in the proper place. 7, -119, 420, -3, 39, and
-120. In
order to find out the inverse of the matrix, all I have to do is dividing 1
by the determinate of the matrix, which is 84 times the transpose of this
matrix. So it will be the transpose of this matrix. It is a C transpose. Let
me write this down. A
inverse is the adjoint of A divided by the
determinate of A and the adjoint of A is nothing
more than the transpose of the cofactor matrix. So the transpose of this
matrix is -4, 7, -3, 80, -119, 39, -384, 420, and -120. That is the transpose
of this matrix. So that is C transpose. And now I divide it by the determinate
of the matrix. And if I do this division I’ll be able to find out what the
inverse of the matrix is, which turns out to be- The determinate of the
matrix is -84- and it turns out to be 0.04672, -0.08333, 0.03571, 0.9524,
1.147, -0.4643, 4.571, -5.000, and 1.429. And that is how we are able to find
the inverse of a matrix. You can extend this procedure to a 4 by 4 or 5 by 5
but the hand computations are going to be lengthy, even if you were going to
use a computer and follow the algorithm here, that
is still going to take a lot of time. And this is the end of this segment. |