ILL 1979 Problem 7

$M = (a_{i,j} ), \ i, j = 1, 2, 3, 4$ , is a square matrix of order four. Given that:

• (i) for each $i = 1, 2, 3,4$ and for each $k = 5, 6, 7$ , $$a_{i,k} = a_{i,k-4};$$$$P_i = a_{1,}i + a_{2,i+1} + a_{3,i+2} + a_{4,i+3};$$$$S_i = a_{4,i }+ a_{3,i+1} + a_{2,i+2} + a_{1,i+3};$$$$L_i = a_{i,1} + a_{i,2} + a_{i,3} + a_{i,4};$$$$C_i = a_{1,i} + a_{2,i} + a_{3,i} + a_{4,i},$$
• (ii) for each $i, j = 1, 2, 3, 4$ , $P_i = P_j , S_i = S_j , L_i = L_j , C_i = C_j$ , and
• (iii) $a_{1,1} = 0, a_{1,2} = 7, a_{2,1} = 11, a_{2,3} = 2$ , and $a_{3,3} = 15$ .

find the matrix M.