Abstract: A set of matrices is said to have the finiteness property if the maximal rate of growth of long products of matrices taken from the set can be obtained by a periodic product. It studies the finite-step realization of the joint/generalized spectral radius of a pair of real square matrices. If S={S_1,S₂}  where S_1,S₂ are  n×n matrices and S₁=(a_ij) with a_ij≥0 and S₂ is similar to a diagonal matrix whose nonzero elements have the same sign, S has the finiteness property.

Key words: the finiteness property; the joint/generalized spectral radius; diagonal matrix