## findme 3 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.

1. Jnlucero

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

2. findme

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.

3. findme

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

4. Jnlucero

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.

5. findme

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

6. Jnlucero

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

7. findme

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

8. findme

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

9. Jnlucero

let me research for it....

10. findme

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