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.
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the characteristic equation is x^2 +x - 6 = 0 so deltaA = -6. thus
A^3 + 2A^2 - 5A + 3i = -216 +72 +30 +3 =-101
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.
I was thinking of Caylay Hamilton Theroem, but I'm not quite familiar with it...
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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.
so you are saying det(A^3 + 2A^2 - 5A + 3i) = (detA)^3 + 2(detA)^2 - 5det(A) +3?
i thought so, since the problem is to find the determinant of the expression.
Is there some kind of theorem to justify what you did? Such as product rule: det(AB) = detA + detB
I simply don't see the justification behind subbing....