I had very good help on this site with my other questions on linear algebra and have one remaining issue with completing my chapter. I worked the problems related to this question OK. The problem I am having is the hint. I do not know if I have posted this on the correct area, please help me figure out where it would go best if I disobeyed the rules for posting.
The problem is this. It is a hint that is assumed to be true but I cannot prove it even though it appears to be true in all cases.
Here it is:
Let L= lambda ( eigenvalue place holder in characteristic equation ) and Ln a specific eigenvalue.
A is an nxn general matrix
Show det(A-LI) = (-1)^n(L-l1)(L-l2)...(L-ln)
Using the factor theorem I can see this is true accept how do I handle the (-1)n ? I also know det(LI-A)= (-1)n(A-LI) but that does not seem to help. The factor theorem will be ture but one of the factors MUST be (ln-L) in the odd case so how can this be proved? Must it be proved some other way.
Any help is appreciated.
Thank you
Julius
The problem is this. It is a hint that is assumed to be true but I cannot prove it even though it appears to be true in all cases.
Here it is:
Let L= lambda ( eigenvalue place holder in characteristic equation ) and Ln a specific eigenvalue.
A is an nxn general matrix
Show det(A-LI) = (-1)^n(L-l1)(L-l2)...(L-ln)
Using the factor theorem I can see this is true accept how do I handle the (-1)n ? I also know det(LI-A)= (-1)n(A-LI) but that does not seem to help. The factor theorem will be ture but one of the factors MUST be (ln-L) in the odd case so how can this be proved? Must it be proved some other way.
Any help is appreciated.
Thank you
Julius