8.10. EXERCISES 191

p(t) and the induction hypothesis. Then plug in t = an and observe the formula isvalid for n.

16. The example in this exercise was shown to me by Marc van Leeuwen and it helped tocorrect a misleading proof of the Cayley Hamilton theorem presented in this chapter.If p(λ ) = q(λ ) for all λ or for all λ large enough where p(λ ) ,q(λ ) are polynomialshaving matrix coefficients, then it is not necessarily the case that p(A) = q(A) for Aa matrix of an appropriate size. The proof in question read as though it was usingthis incorrect argument. Let

E1 =

(1 00 0

),E2 =

(0 00 1

),N =

(0 10 0

)

Show that for all λ ,(λ I +E1)(λ I +E2) =(

λ2 +λ

)I = (λ I +E2)(λ I +E1) . How-

ever,(NI +E1)(NI +E2) ̸= (NI +E2)(NI +E1) .

Explain why this can happen. In the proof of the Cayley-Hamilton theorem givenin the chapter, show that the matrix A does commute with the matrices Ci in thatargument. Hint: Multiply both sides out with N in place of λ . Does N commutewith Ei?

17. Explain why the proof of the Cayley-Hamilton theorem given in this chapter cannotpossibly hold for arbitrary fields of scalars.

18. Suppose A is m×n and B is n×m. Letting I be the identity of the appropriate size,is it the case that det(I +AB) = det(I +BA)? Explain why or why not.

19. Suppose A is a linear transformation and let the characteristic polynomial be

det(λ I−A) =q

∏j=1

φ j (λ )n j

where the φ j (λ ) are irreducible. Explain using Corollary 1.13.10 why the irreduciblefactors of the minimum polynomial are φ j (λ ) and why the minimum polynomial isof the form ∏

qj=1 φ j (λ )

r j where r j ≤ n j. You can use the Cayley Hamilton theoremif you like.

20. M =

B1

. . .

Br

 is a block diagonal matrix. Show det(M) = ∏rk=1 det(Bk).

21. Use the existence of the Jordan canonical form for a linear transformation whoseminimum polynomial factors completely to give a proof of the Cayley Hamilton the-orem which is valid for any field of scalars. Hint: First assume the minimum poly-nomial factors completely into linear factors. In this case, note that the characteristicpolynomial is of degree n and is the product of (λ −µ) where µ is an eigenvalueand listed according to algebraic multiplicity. However, if there are multiple blockscorresponding to some µ, then the minimum polynomial will have such terms but