Prove there exists a matrix

• Jan 10th 2013, 11:04 AM
wilhelm
Prove there exists a matrix
Hi. Here is a problem I found in my algebra book and I don't know how to solve it. Could you please help me?

Show that there exists a matrix $A \in M(n,n;R)$, such that $m_{ij} \in \{-1,0,1\}$ and $det M=1995$

My problem is that I don't know what I should do to prove that there exist a certain matrix.
• Jan 10th 2013, 11:17 AM
Zangeki
Re: Prove there exists a matrix
I may be oversimplifying things, but I think that all you need to do is show a matrix for which this is the case.
You need to find a square matrix (obviously), in which all entries that aren't on the diagonal are either -1, 0 or 1.
This means you could just give a diagonal or triangular matrix, with values on the diagonal that when multiplied together, amount to 1995.
• Jan 10th 2013, 11:22 AM
emakarov
Re: Prove there exists a matrix
Quote:

Originally Posted by Zangeki
You need to find a square matrix (obviously), in which all entries that aren't on the diagonal are either -1, 0 or 1.

Why did you decide that the diagonal elements can be something other than -1, 0 or 1?

Quote:

Originally Posted by Zangeki
This means you could just give a diagonal or triangular matrix, with values on the diagonal that when multiplied together, amount to 1995.

Well, if you multiply -1, 0 and 1, you won't get 1995...

• Jan 10th 2013, 11:24 AM
HallsofIvy
Re: Prove there exists a matrix
Yes, you are over simplifying. The determinant of a diagonal or triangular matrix is just the product of the numbers on the diagonal.
If the numbers in a diagonal or triangular matrix are only -1, 0, or 1, the only possible determinants are -1, 0, and 1 so that is not possible.
• Jan 10th 2013, 11:36 AM
Zangeki
Re: Prove there exists a matrix
Ah, I assumed i couldn't be equal to j. My bad.
• Jan 11th 2013, 04:42 AM
wilhelm
Re: Prove there exists a matrix
I'm sorry. I made a mistake above but I cannot edit the post. There should be detA=1995, not detM
• Jan 11th 2013, 09:24 AM
Opalg
Re: Prove there exists a matrix
I have replied to this problem in another forum.