Question

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.

Answer 1

the characteristic equation is x^2 +x - 6 = 0 so deltaA = -6. thus
A^3 + 2A^2 - 5A + 3i = -216 +72 +30 +3 =-101

Answer 2

Shouldn'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.

Answer 3

I was thinking of Caylay Hamilton Theroem, but I'm not quite familiar with it...

Answer 4

eigenvalues 2 and -3 means the roots of the characteristic equation\[\delta(A-xI)\] 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.

Answer 5

so you are saying det(A^3 + 2A^2 - 5A + 3i) = (detA)^3 + 2(detA)^2 - 5det(A) +3?

Answer 6

i thought so, since the problem is to find the determinant of the expression.

Answer 7

Is there some kind of theorem to justify what you did? Such as product rule: det(AB) = detA + detB

Answer 8

I simply don't see the justification behind subbing....

Answer 9

let me research for it....

Answer 10

oh and you are right about detA = 6, silly me lol