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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 ).