Problem Set 8 (due Friday October 29)
MATH 110: Linear Algebra
Each problem is worth 10 points.
PART 1
1. Curtis p. 201 6.
2. Curtis p. 215 9.
PART 2
Problem 1(20)
a) Prove that if Ais an n×nmatrix with all its eigenvalues equal to λ,
then
exA =eλx
n−1
X
k=0
xk
k!(A−λI)k.
b) A 3 ×3 matrix Ahas all its eigenvalues equal to λ. Show that
exA =1
2eλx((λ2x2−2λx + 2)I+ (−2λx2+ 2x)A+x2A2).
Problem 2(15)
Curtis p. 226 2,3.
Problem 3 (10)
Let Aand Bbe n×nmatrices with detA =detB and trA =trB . Prove
that Aand Bhave the same characteristic polynomial if n= 2, but that this
need not be the case if n > 2.
Problem 4 (10)
Suppose that the minimal polynomial of a linear transformation T:V→
Vsatisfies
m(x) = (x−α1)e
1· · · (x−αs)es
as in the Triangular Form Theorem. Find the rational canonical form of T.
PART 3 - Optional Problem
(Putzer’s Method)
Let λ1, . . . , λnbe the eigenvalues of an n×nmatrix A, and define a
sequence of polynomials in Aas follows: P0(A) = I, and for k= 1, . . . , n:
Pk(A) =
k
Y
m=1
(A−λmI).
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