Linear algebra proofs (linear equations/inverses)

[yankans]

Homework Statement

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?

Homework 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"

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...

 

[Scigatt]

I think #1 is false, actually, unless they mean row-equivalent.

 

[HallsofIvy]

Homework Statement

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?

Homework Equations

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

What is the definition of "equivalent systems"?

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

A matrix is invertible if and only if its determinant is not 0.

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...

 

[Scigatt]

What is the definition of "equivalent systems"?


A matrix is invertible if and only if its determinant is not 0.

The question mentions nothing about square matrices specifically.

 

[HallsofIvy]

The question mentions nothing about square matrices specifically.

? It says:
A = [1 1/2 ... 1/n
1/2 1/3 ...1/(n+1)
...
1/n 1/(n+1)...1/(2n-1)]

Since it has n rows and n columns, that's pretty square!

 

[Scigatt]

? It says:
A = [1 1/2 ... 1/n
1/2 1/3 ...1/(n+1)
...
1/n 1/(n+1)...1/(2n-1)]

Since it has n rows and n columns, that's pretty square!

I was talking about the first part only.

 

[yankans]

The question (1) wants you to prove equivalence in that all the equations in each system are linear combinations of the equations in the other system (I guess that B_11*x_1 + B_12*x_2 +...B_1n*x_n = (c_1*A_11+...+c_m*A_m1)x_1 +(c_1*A_12+...+c_m*A_m2)x_1 +...(c_n*A_1n+...+c_m*A_mn)x_1 = 0).