CHAPTER 05.20: SYSTEM OF
EQUATIONS: Using concept of inverse to solve a set of equations In
this segment we’ll talk about whether we can use the concept of inverse to
solve a set of equations. Let’s suppose somebody gives you an equation of
unknowns. So this is n rows n columns, n rows one column, n
rows one column. And says that hey I’m going to give you the inverse of A so
which means that he or she is telling you that the A
inverse does exist. So the question arises, will it be able to solve for x,
can you find the solution to the set of equations? So let’s see what happens.
If I take A inverse which
I’m assuming which is given to me that it exists. I can multiply both sides
by A inverse. So I get A inverse here like this. But I
know that the A inverse times A is nothing but the identity
matrix. And this one is just A
inverse times C. But then I know that if I multiply the identity matrix by a
vector like this one, so this multiplication is legal, it’s allowed because
the number of columns here is the same as the number of rows here. But I’ll
end up with X itself because anytime a matrix is multiplied by the identity
matrix and that matrix’s multiplication is allowed you end up with the matrix
itself. So that’ll be equal to A
inverse times n by n matrix times C matrix. So what were basically saying is
that hey A inverse, if somebody gives you the
inverse of the caution matrix you should be able to easily find out what the
solution vector is because all you have to do is to write down the inverse
matrix and put the C matrix right here. Multiply the two matrices and all you
have to do is to do the multiplications and you end up with the solution of
the set of equations. So that the good thing about understanding what it
means to find the inverse of a matrix and how to apply it to solve a set of
equations. And this is the end of this segment. |