Hi,

This is probably a simple problem, but I just fell in love with that :D

Prove that for any , the square matrix consisting in only 1's in the diagonal and n everywhere else is always invertible.

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- Jun 30th 2010, 10:25 AMMooInvertible matrix
Hi,

This is probably a simple problem, but I just fell in love with that :D

Prove that for any , the square matrix consisting in only 1's in the diagonal and n everywhere else is always invertible. - Jun 30th 2010, 12:15 PMp0oint
Here is simple proof: The determinant of the matrix will always be hence the matrix will be invertible.

- Jun 30th 2010, 12:16 PMBruno J.
It's not true for ... but for , here goes : it's trivial for , and for , the determinant is equal to , so it can't be zero.

- Jun 30th 2010, 12:16 PMBruno J.
- Jun 30th 2010, 01:04 PMMoo
Yes, sorry, it's only for n>1.

And indeed, the simple way to prove it is to look at the matrix mod n :D

In *hard* times like these, I like algebra (Giggle) - Jun 30th 2010, 01:49 PMBruno J.
Algebra is great! :)

Here's a similar problem, perhaps a little harder.

If is an even positive integer, then the matrix given by , is invertible. - Jun 30th 2010, 10:25 PMsimplependulum
The inverse is :

where and is the size of the matrix .

To obtain the determinant , not only the residue ...

Consider the system of the linear equations :

and note that is your matrix , let it be ( If i read the problem correctly )

Let be the quantity then we have

So

I believe it is true that - Jul 1st 2010, 01:49 AMsimplependulum
- Jul 1st 2010, 02:51 AMtonio
- Jul 1st 2010, 03:49 AMsimplependulum
Oh , the matrix is based on what i defined in the previous post (#7) , sorry .