In this manuscript we introduce the class of β-matrices, which gives a new sufficient condition for the positivity of the determinant. However, we show that nonnegative β-matrices are not necessarily P-matrices. For column stochastic matrices, the property of being a β-matrix is weaker than strict diagonal dominance. We extend β-matrices to tensors and call them β-tensors. Although they are not in general P-tensors, we prove that nonnegative β-tensors of odd order are P-tensors.