An
![n \times n](/media/m/9/d/8/9d8eac5b3234425afb9f970edbfe93ef.png)
matrix whose entries come from the set
![S = \{1, 2, \ldots , 2n - 1\}](/media/m/d/e/5/de502c52424b33902f262ec149c18556.png)
is called a silver matrix if, for each
![i = 1, 2, \ldots , n](/media/m/0/1/c/01cc0db37f6db9a31d132b248a2149d8.png)
, the
![i](/media/m/3/2/d/32d270270062c6863fe475c6a99da9fc.png)
-th row and the
![i](/media/m/3/2/d/32d270270062c6863fe475c6a99da9fc.png)
-th column together contain all elements of
![S](/media/m/c/6/3/c63593c3ec0773fa38c2659e08119a75.png)
. Show that:
(a) there is no silver matrix for
![n = 1997](/media/m/5/7/6/576cebbdbccabc4a21c27da44ccde2f1.png)
;
(b) silver matrices exist for infinitely many values of
![n](/media/m/a/e/5/ae594d7d1e46f4b979494cf8a815232b.png)
.
%V0
An $n \times n$ matrix whose entries come from the set $S = \{1, 2, \ldots , 2n - 1\}$ is called a silver matrix if, for each $i = 1, 2, \ldots , n$, the $i$-th row and the $i$-th column together contain all elements of $S$. Show that:
(a) there is no silver matrix for $n = 1997$;
(b) silver matrices exist for infinitely many values of $n$.