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If A = (aij) = M3 (K) for a field K then the characteristic polynomial of A has the form p₁(x) = - det(A) + C(A)x − Tr(A)x² + x³ where the coefficient of a can be computed as C(A) = a11@22 + a22a33 + a11933 a12a21a13a31 - a23a32 (a) is it necessarily true that if A is invertible then C(A¯¹) = C(A)? (b) is it necessarily true that if det (A) = Tr(A) = 0 then A³ = -C(A)A?

If A = (aij) = M3 (K) for a field K then the characteristic polynomial of A has the form p₁(x) = - det(A) + C(A)x − Tr(A)x² + x³ where the coefficient of a can be computed as C(A) = a11@22 + a22a33 + a11933 a12a21a13a31 - a23a32 (a) is it necessarily true that if A is invertible then C(A¯¹) = C(A)? (b) is it necessarily true that if det (A) = Tr(A) = 0 then A³ = -C(A)A?

Elements Of Modern Algebra
Elements Of Modern Algebra
8th Edition
ISBN:9781285463230
Author:Gilbert, Linda, Jimmie
Publisher:Gilbert, Linda, Jimmie
Chapter8: Polynomials
Section8.3: Factorization In F [x]
Problem 13E
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If A = (aij) = M3 (K) for a field K then the characteristic polynomial of A has the form p₁(x) = - det(A) + C(A)x − Tr(A)x² + x³ where the coefficient of a can be computed as C(A) = a11@22 + a22a33 + a11933 a12a21a13a31 - a23a32 (a) is it necessarily true that if A is invertible then C(A¯¹) = C(A)? (b) is it necessarily true that if det (A) = Tr(A) = 0 then A³ = -C(A)A?
Transcribed Image Text:If A = (aij) = M3 (K) for a field K then the characteristic polynomial of A has the form p₁(x) = - det(A) + C(A)x − Tr(A)x² + x³ where the coefficient of a can be computed as C(A) = a11@22 + a22a33 + a11933 a12a21a13a31 - a23a32 (a) is it necessarily true that if A is invertible then C(A¯¹) = C(A)? (b) is it necessarily true that if det (A) = Tr(A) = 0 then A³ = -C(A)A?
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