Solving for Eigenvalues and Determinant of a Singular Matrix

[daniel_i_l]

Homework Statement

You have a singular 3x3 matrix where det(A-I) = 0 , rank(A+3I) = 2
a) find the characteristic polynomial of A.
b) Is A-3I invertible?
c) Find det(A) and tr(A)

Homework Equations

The Attempt at a Solution

a) First I'll find the eigenvalues:
- 0 is obviously one since A is singular
-det(A-I)=0 => det(I-A) = 0 and so 1 is also an eigenvalue.
-rank(A+3I)=2 => rank(-3I - A) = 2 and so det(-3I - A) = 0 and so -3 is also one.
So the polynomial is t(t-1)(t+3)

b) No because if det(A-3I)=0 => det(3I-A)=0 => 3 is an eigenvalue but we already found 3 and that's the maximum.

c) det(A) =0 , and the polynomial is t(t^2+2t -3) = t^3 +2t^2 -3t and so tr(A) = 2.

Is that right? Am I missing anything?
Thanks.

 

[Dick]

Basically fine, except that you argument in b) shows A-3I IS invertible, correct? And an alternative way to get the trace is just to sum the eigenvalues 0+1-3=-2. So since the characteristic polynomial is (t-e1)(t-e2)(t-e3) you can see that the coefficient of the t^2 is NEGATIVE the sum of the eigenvalues.

 

[Mindscrape]

For part b:

Do you know the theorem that says that if you have any zero eigenvalue then the matrix is singular? If not, does it make sense to you why this is true?

 

[daniel_i_l]

Thanks everyone.