The characteristic polynomial of a 5 × 5 matrix is given below. Find the eigenvalues and their multiplicities.
-$\lambda ^ { 5 } - 24 \lambda ^ { 4 } + 189 \lambda ^ { 3 } - 486 \lambda ^ { 2 }$
A) 0 (multiplicity 2), 9 (multiplicity 2), 6 (multiplicity 1 )
B) 0 (multiplicity 2), $- 9$ (multiplicity 2), -6 (multiplicity 1)
C) 0 (multiplicity 2), $- 9$ (multiplicity 2), 6 (multiplicity 1 )
D) 0 (multiplicity 1), 9 (multiplicity $3 ), 6$ (multiplicity 1 )

Question

Quizplus · Accepted Answer

The characteristic polynomial is missing its constant term, which implies it is divisible by $\lambda^2$, indicating $\lambda = 0$ has multiplicity 2. Factoring the polynomial as $\lambda^2(\lambda^3 - 24\lambda^2 + 189\lambda - 486)$ and solving for the roots of the cubic part will give the other eigenvalues. The correct factorization of the cubic part gives $\lambda = 9$ (multiplicity 2) and $\lambda = 6$ (multiplicity 1), making option A correct.

The Characteristic Polynomial of a 5 × 5 Matrix Is λ5−24λ4+189λ3−486λ2\lambda ^ { 5 } - 24 \lambda ^ { 4 } + 189 \lambda ^ { 3 } - 486 \lambda ^ { 2 }λ5−24λ4+189λ3−486λ2

The Characteristic Polynomial of a 5 × 5 Matrix Is $\lambda ^ { 5 } - 24 \lambda ^ { 4 } + 189 \lambda ^ { 3 } - 486 \lambda ^ { 2 }$