A community for students. Sign up today!
Here's the question you clicked on:
 0 viewing
 2 years ago
Let A be a 2x2 matrix whose eigenvalues are 2 and 3. What is the determinant of A^3 + 2A^2 5A + 3i, where i is the identity matrix.
 2 years ago
Let A be a 2x2 matrix whose eigenvalues are 2 and 3. What is the determinant of A^3 + 2A^2 5A + 3i, where i is the identity matrix.

This Question is Closed

Jnlucero
 2 years ago
Best ResponseYou've already chosen the best response.0the characteristic equation is x^2 +x  6 = 0 so deltaA = 6. thus A^3 + 2A^2  5A + 3i = 216 +72 +30 +3 =101

findme
 2 years ago
Best ResponseYou've already chosen the best response.0Shouldn't the characteristic equation be x^2 + x + 6 =0 so detA = 6? anyhow, I don't understand how you can just sub detA into the equation to the the determinant of A^3 + 2A^2  5A + 3i, please explain.

findme
 2 years ago
Best ResponseYou've already chosen the best response.0I was thinking of Caylay Hamilton Theroem, but I'm not quite familiar with it...

Jnlucero
 2 years ago
Best ResponseYou've already chosen the best response.0eigenvalues 2 and 3 means the roots of the characteristic equation\[\delta(AxI)\] are 2 and 3. this gives x^2 + x 6 = 0 since r +s = 1 and rs = 6. thus \[\delta(A)\] is the constant term 6. thus substitution gives the answer.

findme
 2 years ago
Best ResponseYou've already chosen the best response.0so you are saying det(A^3 + 2A^2  5A + 3i) = (detA)^3 + 2(detA)^2  5det(A) +3?

Jnlucero
 2 years ago
Best ResponseYou've already chosen the best response.0i thought so, since the problem is to find the determinant of the expression.

findme
 2 years ago
Best ResponseYou've already chosen the best response.0Is there some kind of theorem to justify what you did? Such as product rule: det(AB) = detA + detB

findme
 2 years ago
Best ResponseYou've already chosen the best response.0I simply don't see the justification behind subbing....

Jnlucero
 2 years ago
Best ResponseYou've already chosen the best response.0let me research for it....

findme
 2 years ago
Best ResponseYou've already chosen the best response.0oh and you are right about detA = 6, silly me lol
Ask your own question
Ask a QuestionFind more explanations on OpenStudy
Your question is ready. Sign up for free to start getting answers.
spraguer
(Moderator)
5
→ View Detailed Profile
is replying to Can someone tell me what button the professor is hitting...
23
 Teamwork 19 Teammate
 Problem Solving 19 Hero
 Engagement 19 Mad Hatter
 You have blocked this person.
 ✔ You're a fan Checking fan status...
Thanks for being so helpful in mathematics. If you are getting quality help, make sure you spread the word about OpenStudy.