Is Every Eigenvalue Claim Correct for Matrix Operations?

[shiri]

Let A and B be nxn matrices, where B is invertible. Suppose that 4 is an eigenvalue of A, and 5 is an eigenvalue of B. Find ALL true statements.

A) 4 is an eigenvalue of A^T
B) 4 is an eigenvalue of (B^−1)AB
C) 265 is an eigenvalue of (A^4)+A+5I
D) 8 is an eigenvalue of A+(A^T)
E) 20 is an eigenvalue of AB
F) None of the above

I choose:
A)
B)

am I right?

 

[omega101]

Let A and B be nn matrices, where B is invertible. Suppose that 5 is an eigenvalue of A, and 4 is an eigenvalue of B. Find ALL true statements below.
A. 20 is an eigenvalue of AB
B. 10 is an eigenvalue of A+AT
C. 34 is an eigenvalue of A2+A+4I
D. 5 is an eigenvalue of AT
E. 5 is an eigenvalue of B−1AB
F. None of the above

HELP!

 

[shiri]

Let A and B be nxn matrices, where B is invertible. Suppose that 4 is an eigenvalue of A, and 5 is an eigenvalue of B. Find ALL true statements.

A) 4 is an eigenvalue of A^T
B) 4 is an eigenvalue of (B^−1)AB
C) 265 is an eigenvalue of (A^4)+A+5I
D) 8 is an eigenvalue of A+(A^T)
E) 20 is an eigenvalue of AB
F) None of the above

I choose:
A)
B)

am I right?

anyone?

 

[Dick]

If you think a statement is true you should probably give a reason. If you think it's not true then you should figure out a counterexample. Otherwise you are just playing a guessing game.

 

[shiri]

If you think a statement is true you should probably give a reason. If you think it's not true then you should figure out a counterexample. Otherwise you are just playing a guessing game.

A) Say it was a 3x3 matrix. If I transpose the matrix, the eigenvalue still be 4.
B) BB^-1 becomes an identity and it lefts with A only.
C) A^4 + A + 5I = (4)^4 + 4 + 5 = 265. However, I'm not too sure about this statement.
D) Not too sure if I can simply add those two together. If I assume that upper and lower triangular part are zeros.
E) Obviously not a correct statement.

What do you think, Dick? A & B only?

I am very skeptical about C & D. Please help me on this, Dick.

 

[Dick]

Ok, for A. A better reason is that the eigenvalues are the roots of det(A-xI) and determinant of the transpose of A-xI is the same as the determinant of A-xI. Your reason for B isn't so good. Matrices in general don't commute. If v is the eigenvector of A then Av=4v. Define a vector y=B^(-1)x. What's (B^(-1)AB)y? For C, use Av=4v and figure out what (A^4+A+5I)v is. What does that tell you about the eigenvalues of (A^4+A+5I)? Can you think of a counterexample for D and E?

 

[shiri]

Ok, for A. A better reason is that the eigenvalues are the roots of det(A-xI) and determinant of the transpose of A-xI is the same as the determinant of A-xI. Your reason for B isn't so good. Matrices in general don't commute. If v is the eigenvector of A then Av=4v. Define a vector y=B^(-1)x. What's (B^(-1)AB)y? For C, use Av=4v and figure out what (A^4+A+5I)v is. What does that tell you about the eigenvalues of (A^4+A+5I)? Can you think of a counterexample for D and E?

A, B, C are true

D) If I assume there are non-zeros in the matrix

eg.
|1 4|+|1 3|=|2 7|
|3 2| |4 2| |7 4|

So, eval are roots of different roots and I have to use a quadratic formula to find those roots

E) Multiplying A&B will not come out as 20 b/c the matrix should be multiplied, not multiplying scalars (eigenvalues). So, False.

What do you think?

 

[Dick]

Yes, A, B and C are true. D and E are only true for some matrices, not for others. I'll give you examples where they aren't and you figure out why, ok? I can't really tell what you are trying to say for either of them. For D take A=[[4,1],[0,0]]. What are the eigenvalues of A, A^(T) and A+A^(T)? For E take A=[[4,0],[0,0]] and B=[[0,0],[0,5]]. What are the eigenvalues of A, B and AB? If you want another exercise, figure out specific matrices where D and E ARE true.