IMO 1973 SL 12

Origin: SWE

Problem

Consider the two square matrices A = ⎡ ⎢⎢⎢⎢⎣ 1 −1 −1 1 −1 −1 1 −1 −1 −1 1 −1 1 −1 ⎤ ⎥⎥⎥⎥⎦ and B = ⎡ ⎢⎢⎢⎢⎣ 1 −1 −1 1 −1 1 −1 1 −1 −1 1 −1 1 −1 ⎤ ⎥⎥⎥⎥⎦ with entries 1 and −1. The following operations will be called elementary: (1) Changing signs of all numbers in one row; (2) Changing signs of all numbers in one column; (3) Interchanging two rows (two rows exchange their positions); (4) Interchanging two columns. Prove that the matrix B cannot be obtained from the matrix A using these operations.

Solution

Observe that the absolute values of the determinants of the given matrices are invariant under all the admitted operations. The statement follows from det A = 16 ̸= det B = 0.