# Linear algebra proofs (linear equations/inverses)

1. Feb 9, 2010

### yankans

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...

2. Feb 14, 2010

### Scigatt

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

3. Feb 14, 2010

### HallsofIvy

Staff Emeritus
What is the definition of "equivalent systems"?

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

4. Feb 15, 2010

### Scigatt

The question mentions nothing about square matrices specifically.

5. Feb 15, 2010

### HallsofIvy

Staff Emeritus
??? 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!

6. Feb 15, 2010

### Scigatt

I was talking about the first part only.

7. Feb 16, 2010

### 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).