anonymous
  • anonymous
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.
Mathematics
  • Stacey Warren - Expert brainly.com
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SOLVED
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jamiebookeater
  • jamiebookeater
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anonymous
  • anonymous
the characteristic equation is x^2 +x - 6 = 0 so deltaA = -6. thus A^3 + 2A^2 - 5A + 3i = -216 +72 +30 +3 =-101
anonymous
  • anonymous
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.
anonymous
  • anonymous
I was thinking of Caylay Hamilton Theroem, but I'm not quite familiar with it...

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anonymous
  • anonymous
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.
anonymous
  • anonymous
so you are saying det(A^3 + 2A^2 - 5A + 3i) = (detA)^3 + 2(detA)^2 - 5det(A) +3?
anonymous
  • anonymous
i thought so, since the problem is to find the determinant of the expression.
anonymous
  • anonymous
Is there some kind of theorem to justify what you did? Such as product rule: det(AB) = detA + detB
anonymous
  • anonymous
I simply don't see the justification behind subbing....
anonymous
  • anonymous
let me research for it....
anonymous
  • anonymous
oh and you are right about detA = 6, silly me lol

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