Problem 4

n\times n

matrix whose entries belong to

S=\{1,2,\ldots,2n-1\}

is called a silver matrix if, for each

i

, the

i

th row together with the

i

th column contains all elements of

S

. Show that (a) there is no silver matrix for

n=1997

; (b) silver matrices exist for infinitely many

n

Step 3 of 4: Define the doubling blocks

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In plain words

Define the doubling blocks

B_{i,j}=A_{i,j}

Detailed analysis

Given a silver matrix $A$ of size $n$ with diagonal entries $1$ , form a $2n$ matrix by $B_{i,j}=A_{i,j}$ , $B_{i+n,j+n}=A_{i,j}$ , $B_{i,j+n}=2n+A_{i,j}$ , and $B_{i+n,j}=2n+A_{i,j}$ for $i\ne j$ , while $B_{i+n,i}=2n$ .