SELECTED SOLUTIONS FROM THE HOMEWORK
ANDREW J. BLUMBERG
1. Solutions
(2.4, 18) Assume that A and B are nonsingular n × n matrices. Prove that A and
B commute if and only if (AB)2 = A2 B 2 .
Proof. If A and B commute, then
(AB)2 = (AB)(AB) = A(BA)B = A(AB)B = A2 B 2 .
On the other hand, if ABAB = A2 B 2 , left multiplying by A−1 yields the
expression BAB = AB 2 , and right multiplying by B −1 leaves us with
BA = AB.
(2.4, 20) Prove that, if A is an n × n matrix and A − I is nonsingular, then for every
integer k ≥ 0,
I + A + A2 + . . . + Ak = (Ak+1 − I)(A − I)−1
Proof. This follows by computing out (I + A + . . . + Ak )(A − I) and then
multiplying by (A − I)−1 .
(3.2, 7) Suppose that AB = AC and det(A) 6= 0. Show that B = C.
Proof. Since det(A) 6= 0, A−1 exists. Left multiplying AB = AC by A−1
then implies that B = C.
E-mail address: [email protected]
Date: March 9, 2015.
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