# Non-zero diagonal through permutation of rows

I'm looking for a reference of the following statement (which can easily be proved by Laplace's formula and induction):

Let $R$ be a commutative ring with identity and let $A$ be an invertible matrix over $R$. Then there is a permutation matrix $P$ such that the diagonal of $PA$ has no zero.

Edit: Of course, $R$ has to be a domain to make the arguments (formula of Leibniz or Laplace) work. Futhermore, it's sufficient to require $det(A) \neq 0$ (instead of $A$ being invertible).

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I'm hoping you won't find a reference of that statement, since it is false. (Consider the 1x1 matrix over the zero ring.) –  Ricky Demer May 5 '11 at 21:13
Doesn't a ring with identity have at least two elements? –  Tony Huynh May 5 '11 at 21:19
No, Tony. Otherwise algebra would be swamped in statements like "if $I$ is an ideal of $R$, then $R/I$ is a ring or zero". –  darij grinberg May 5 '11 at 21:21
Makes sense. Thanks darji. –  Tony Huynh May 5 '11 at 21:26
I don't know of a reference, but you can also appeal to the determinant formula as a sum of (permuted) products for a shorter proof. Then you may not need a reference. Gerhard "Ask Me About System Design" Paseman, 2011.05.05 –  Gerhard Paseman May 5 '11 at 22:08

## 1 Answer

There is a strictly stronger fact about matrices:

Given $A\in{\bf M}_n(R)$, the following statements are equivalent.

• There exists a permutation matrix $P$ such that the diagonal of $PA$ has no zero.
• If an $m\times p$-block of $A$ is mades of zeroes, then $m+p\le n$.
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And it's called Hall's marriage theorem. –  darij grinberg May 6 '11 at 19:38
@darij, I'd rather say it can be proved using Hall's Marriage Theorem, or any of a number of other theorems equivalent to Hall's, including one, by Konig, that goes back at least 20 years before Hall, and was stated specifically in the context of matrices. –  Gerry Myerson May 6 '11 at 22:55