CHAPTER 05.21: SYSTEM OF
EQUATIONS: Finding the inverse of a matrix Theory In
this segment we値l talk about how we will find the inverse for a matrix. If A
is a n by n matrix, so if we have a square matrix
and we claim that A inverse is the inverse of A,
then we know that hey A times A inverse will be
equal to the identity matrix. So the question arises that how do we find out
the inverse of a matrix so at least get an idea of how we should go about
doing that. So if we consider the A matrix, let痴 suppose for a n by n matrix to be of this form, so we have A one one going all the way to A one n. A
n one going all the way to A and n so we have these n by n elements, n
squared elements in the A matrix. And then if we have the A inverse matrix
and that also has n square elements let痴 call it A
one one prime all the way up to A one n prime. Then
A n one prime and A nn
prime. Then
how will we be able to find what the inverse of a matrix is? All we have to
do is to multiply this matrix by this matrix and put it into the identity
matrix. So that痴 how we値l be able to do that. So if we have this matrix
here we have A one one, all the way up to A one n
then A n one all the way up to A nn I am going to
multiply it by the inverse of a matrix which will be A one one prime, A one n
prime, A n one prime all the way up to A nn prime.
And I知 going to put that equal to the identity matrix. So I get one, zero,
all the way to zero, zero, one, zero and all the way to zero here and one
here. So what I basically have is that our n by n matrix which I知
multiplying by its inverse and I知 supposed to get the identity matrix right
here. So the question arises: how do I find out what the inverse of the
matrix. What I can look at at this phase is that
now I can now rewrite finding the inverse of the matrix as a system of
simultaneous linear equations as follows. I can say hey
let me take the first column of the inverse matrix. So if I take the first
column of the inverse matrix and I multiply it to the A matrix, it値l
basically give me the first column of the identity matrix. Which makes sense
because if I take the first row here and first column here, I値l get the
first row first column here. If I take the first row here, second row here
and first column here, I値l get the second row first column here. If I take
the nth row here and the first column here I値l get the nth row first column
here. So what that痴 going to do is that if I now write down only the first
column here of the inverse matrix then ill only get a column matrix on the
right side also. So this is A one one and it goes to the A one n, and this is A n one and
that痴 A nn I値l write this as A
one prime all the way up to A n one prime. So that痴 my first column of the
inverse of the matrix. Then this one will just be one zero, zero, zero. So
what that means is in order to find the first column of the inverse of the
matrix, I値l solve n equations and unknowns which will be of this particular
matrix form. And
what does that mean in terms of finding the other columns of the inverse of
the matrix? If I would have put the second column of the inverse of the
matrix here, then in this case the right hand side would become zero one zero
zero zero so on and so
forth. And so what that means is that I値l have the second column of the identity
matrix would be my right hand side vector. So if I continue doing that I値l
be able to set up an equation n unknowns but our n such sets of n equations n
unknowns and I値l be able to find the inverse of the matrix. And that is the
end of this segment. |