Question

The characteristic polynomial of a 5x5 matrix is given below. Find the eigenvalues and their multiplicities.

λ5-11λ4+18λ3

I know the answer is 0(multiplicity 3), 2(multiplicity 1), 9 (multiplicity 1)

but I have absolutely no idea how to get the answer. A question similiar to this will be on my test. Please show steps to solve. I often get lost in the vocab.

Answer 1

Explanation:

given that the characteristic polynomial of a 5x5 matrix is given below.

`\lambda ^5-11\lambda ^4+18\lambda ^3`

The solution is simple

Equate the above 5th degree polynomial in lambda to 0 and solve for lambda

We would have 5 solutions

Equating to 0 and factorising we get

`=\lambda ^3(\lambda ^2-11\lambda +18)\\= \lambda ^3(\lambda -9)(\lambda -2)=0`

Equate each factor to 0

Since lambda has power 3 , we say that 0 is a root with multiplicity 3

(this means 3 times same value repeated as root)

Other roots are 9 and 1 with multiplicity 1 each.