  The following theorem is proved:
  Suppose $M = (a_{i,j})$ be a $k \times k$ matrix with positive entries and
$a_{i,j}a_{i+1,j+1} > 4\cos ^2 \frac{\pi}{k+1} a_{i,j+1}a_{i+1,j} \quad (1 \leq
i \leq k-1, 1 \leq j \leq k-1).$ Then $\det M > 0 .$ The constant $4\cos ^2
\frac{\pi}{k+1}$ in this Theorem is sharp. A few other results concerning
totally positive and multiply positive matrices are obtained.
  Keywords: Multiply positive matrix; Totally positive matrix; Strictly totally
positive matrix; Toeplitz matrix; Hankel matrix; P\'olya frequency sequence.
