(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Two problems.

(1) Prove that if two homogeneous systems of linear equations in two unknowns have the same solutions, then they are equivalent.

(2) Can you prove that the matrix

A = [1 1/2 ... 1/n

1/2 1/3 ...1/(n+1)

...

1/n 1/(n+1)...1/(2n-1)]

is invertible and that A^(-1) has integer entries?

2. Relevant equations

(1) Perhaps the theorem - Equivalent systems of linear equations have the same solutions? It is the other way round.

(2)hmm...I don't know of "relevant equations"

3. The attempt at a solution

Not very sure how to start. For the first one, tried to construct arbitrary matrices A and B to represent two different homogeneous systems, but didn't get very far.

For (2), tried row-reducing the matrix > echelons but don't know how a proof will result...

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# Linear algebra proofs (linear equations/inverses)

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