CHAPTER 05.21: SYSTEM OF
EQUATIONS: Finding the inverse of a matrix Example In
this segment we will look at how to find the inverse of matrix through an
example. Let’s suppose somebody says hey go head and find out the inverse of
this matrix 25, 5, 1, 64, 8, 1, 144, 12, 1 so I will see from the definition
of the inverse of a matrix if I multiply this square matrix by its inverse I
will get the identity matrix of course we are assuming here that the inverse
exists. So we’re not looking into that if we face any problems in finding the
solution to the set of equations then we know that hey we have some problems
with finding the inverse of the matrix, but let’s suppose in this case it is
already given to you that the inverse exists so this will be a a 1 1 prime, a 1 2 prime, a 1 3 prime, a 2 1 prime, a 2 2
prime, a 2 3 prime, so on and so forth. So this is the matrix that you should
find for an inverse this is the inverse of the matrix so this is [A] this is
[A] inverse and this will be the identity matrix 1, 0, 0, 0, 1, 0, 0, 0, 1. So
if we’re going to solve for the inverse of the matrix what we can do is we
can only look at the first column let’s suppose of the inverse matrix, then
the answer will be the first column here. If we want to find out the second
column of the inverse of matrix then your answer will be the second column of
the identity matrix; if I want to find the third column of the inverse of a
matrix I will use the third column of the identity matrix as my right hand
side by able to doing that I will be able to set up my equation. So let’s go
ahead and see how that turns out to be the case. So what we have is we have
25, 5, 1, 64, 8, 1, 144, 12, 1 and then the first column of the inverse
matrix is a 1 1 prime, a 2 1 prime, and a 3 1 prime and that’ll be equal to
first column of the identity matrix which is 1, 0, 0. So if that’s the case
all we have to do is solve these three equations three unknowns and we are
not going to show you how to solve the three equations three unknowns because
it’s not the best way to solve the inverse of a matrix. But we want to just
get to the concept of what it means to have the inverse of a matrix and how
to find it. So
if I solve this set of equations by any of the methods which you might have
learned so far I will get the four way solution from the matrix from the set
of equations being solved I’ll get 0.04762, - 0.9524, 4.571.
So this basically tells me - hey this is the first column of the inverse of
the matrix. Now we’re going to find out the second column of the inverse
matrix so we have 25, 5, 1, 64, 8, 1, 144, 12, 1 and I’m going to put the
second column of the inverse matrix here, which is a 1 2 prime, a 2 2 prime,
and a 3 2 prime. This is the second column of the inverse matrix and that’ll
be equal to the right hand side which is 0, 1, 0, which is the second column
of the identity matrix. Again we want to solve these three equations three
unknowns and this is what we’re going to get as our solution for our second
column of the inverse matrix -0.08333, 1.417 and -5.000. So that’s what I’m
going to get as the second column of the inverse of the matrix. Let’s
go and see how we can find the third column of the inverse of the matrix,
which we have 25, 5, 1, 64, 8, 1, 144, 12, 1; and now the third column of the
inverse matrix would be as follows - would be a 1 3 prime, a 2 3 prime, a 3 3
prime. I’m going to do 0, 0, 1 right here so in order to find the third
column of the inverse of this matrix, I will choose the third column of the identity
matrix as my right hand side and solve these three equations three unknowns
and I’ll be able to find out what my third column of the inverse of the
matrix is. We
are not talking about how we would solve these three equations three unknowns because
you can use any method you want to and also this is not the best way to find
the inverse of the matrix, but at this time we are just interested in
reporting how what it means to find the inverse of a matrix. So in this case
we get 0.03571 as the element there, -0.4643 right here and 1.429 here. So
that’s our third column so what we’re going to do now is that we’re going to
write down all the three columns which we have obtained so far. So the
inverse of the matrix which we have found will be the first column which we
found which we got as 0.04762, - 0.9524, 4.571 and then the second column is
-0.08333, 1.417 and -5.000 and the third column which we get which we have
just written here will be 0.03571, -0.4643, and 1.429. So that’s the inverse
of the [A] matrix; so if you would go ahead as an exercise what you can do is
take the [A] matrix and the [A] inverse matrix and verify that when you
multiply the inverse matrix by [A] that you are going to get the identity
matrix. And that’s the end of this segment. |