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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.
 one year ago
 one year 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.
 one year ago
 one year ago

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

findmeBest ResponseYou've already chosen the best response.0
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.
 one year ago

findmeBest ResponseYou've already chosen the best response.0
I was thinking of Caylay Hamilton Theroem, but I'm not quite familiar with it...
 one year ago

JnluceroBest ResponseYou've already chosen the best response.0
eigenvalues 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.
 one year ago

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

JnluceroBest ResponseYou've already chosen the best response.0
i thought so, since the problem is to find the determinant of the expression.
 one year ago

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

findmeBest ResponseYou've already chosen the best response.0
I simply don't see the justification behind subbing....
 one year ago

JnluceroBest ResponseYou've already chosen the best response.0
let me research for it....
 one year ago

findmeBest ResponseYou've already chosen the best response.0
oh and you are right about detA = 6, silly me lol
 one year ago
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