1) Given a matrix M in RREF, we refer to the positions of its leading ones as the leading one configuration of M

1) Given a matrix M in RREF, we refer to the positions of its leading ones as the leading one configuration of M. For example, there are exactly 4 distinct leading one configurations for

2 x 2 matrices, namely

Here we denote by * an entry that can take any real value.

(a) Write down the list of distinct leading one configurations for 2 x 3 matrices. Explain briefly but clearly why your list is complete.

(b) Write down the list of distinct leading one configurations for 3 x 3 matrices.

(c) Find (in terms of n) the number of distinct leading one configurations for 2 x n matrices, where n > 2. Prove that your answer is correct (preferably using induction).

(d) Find (in terms of n) the number of distinct leading one configurations for n x n matrices, where n > 2. No proof required for this part.

2) Let M_n´n denote the set of n x n matrices.

(a) If A Î M_n´n is invertible and X Î M_n´n is such that AX = 0, show that X = 0.

(b) If A Î M_n´n is not invertible, show that there exists a nonzero X Î M_n´n such that AX = 0.

(c) Let A Î M_n´n be fixed and consider the function ¦ : M_n´n ® M_n´n defined by

f(X) = AX

Prove that ¦ is injective if and only if A is an invertible matrix.

(d) Fix an integer n > 1 and consider the function g : M_n´n ® R defined by

g(X) = (det x)²⁰²¹

Is g injective? Surjective? Bijective? Explain your answers.

3) Consider the following subset of N:

S := {6n-—1|n Î N} = {5, 11,17, 23, 29, 35,...}

(a) Show that every s Î S must be divisible by some prime p Î S. (You may assume the Fundamental Theorem of Arithmetic is true.)

(b) Consider the following proof of the statement “The set S contains infinitely many prime numbers ”:

Proof. Suppose S' contains only finitely many prime numbers, and list all of them:

P₁ = 6n₁ —1, P₂ = 6n₂ —1, ..., p_k = 6n_k — 1

Define the number

S = 6p₁p₂...P_k —1

Clearly s Î S, and by part (a) it must be divisible by some prime p Î S. But s is not divisible by p₁,p₂,...,p_k and these are all the primes in S, contradiction. L

i. Is the statement “The set S contains infinitely many prime numbers” true if we replace 6n — 1 in the definition of S by 6n + 2? Or 6n + 3? Or 6n+ 4? Explain your reasoning.

li. ‘Try to modify the proof given above to prove the statement “The set T’ contains infinitely many prime numbers”, where

T := {6n+1|n Î N} = {7,13,19, 25,...}

Discuss your success or lack thereof.