HOMEWORK 6.
Due Monday, October 24, in class.
1. A matrix A ∈ Mn×n (R) is a 01-matrix if Aij ∈ {0, 1}, for all 1 ≤ i, j ≤ n.
What is the largest number of 1’s that an invertible 01-matrix of size n × n can
have?

2. Suppose that a matrix Mn×n (F) can be written in the form M = 
A
B

,
C D
where A and D are square matrices. Prove that if C and D commute, and if D is
invertible, then a necessary and sufficient condition that M be invertible is that
AD − BC be invertible. Construct examples to show that if the assumption
that D is invertible is dropped, then the condition becomes unnecessary and
insufficient.
3. Do Problem 22 (c) in Section 4.3.
4. Let A ∈ Mn×n (R) and suppose that
Prove or disprove: det(A) > 0.
P
j6=i
|Aij | < Aii for each i, 1 ≤ i ≤ n.
5. Let V = Mn×n (F), let B ∈ V , and let TB ∈ L(V ) be defined by TB (A) =
AB − BA. Calculate det(TB ).