Problem Set 2
due at 5pm on April 01, 2023
Directions: Justify all answers, show all solutions, and work independently. To get full credit,
proofs and explanations should be clear, coherent, complete, and concise; and examples should be
verified to satisfy prescribed properties. (Total: 10 pts.)
1. (a) Let A, C ∈ Mn (F) and r ∈ F.
Show that if A is similar to C, then (A − rI)k is similar to (C − rI)k for all k ∈ Z+ .
#
#
"
"
r 0 3
r 3 −2
(b) Let A = 0 r 0 , C = 0 r −2 ∈ M3 (R).
0 0 r
0 0 r
i. Are A and C equivalent and why?
ii. Are A and C similar? If no, explain why. If yes, find nonsingular P ∈ M3 (R) such
that A = P CP −1 .
2. Let A ∈ Mm,n (F). Show that
rankA = 1 if and only if there exist nonzero x ∈ Mm,1 (F) and y ∈ Mn,1 (F) such that A = xy > .
3. Let b, a1 , . . . , an ∈ F such that b 6= 0. Prove that


b
0 0 ...
0
a1
b 0 ...
0
a2 
 −1
 0 −1
b ...
0
a3 



 . .
.
.
.
.
 .. . . . . . .
..
..  ∈ Mn (F)




.
 0
0 . . −1
b
an−1 
0
0 ...
0 −1 b + an
has determinant bn + an bn−1 + an−1 bn−2 + . . . + a2 b + a1 .
4. Let A ∈ Mn (R) be nonsingular such that all entries are integers.
Show that all entries of A−1 are integers if and only if detA = ±1.
5. (a) Let A ∈ Mm,n (F). Use the theory developed in class to explain why rankA = rankA> ,
i.e., the maximal number of linearly independent columns of A (viewed as vectors in
Mm,1 (F)) is equal to the maximal number of linearly independent rows of A (viewed as
vectors in M1,n (F)).
(b) Show that if A ∈ Mn (F) and rankA = n − 1, then adjA has a nonzero entry and
rank(adjA) = 1.
0n −2In
6. If A =
∈ M2n (C), determine the characteristic and minimal polynomials of A.
In
In