<p>The chainable index, as well as the scrambling index and exponent, is a combinatorial invariant of nonnegative matrices, which is important in describing their properties. In this paper, the linear bijective maps that preserve either the smallest or the largest value of the chainable index are described. As a corollary, a characterization of the linear bijective maps that preserve all values of the chainable index is obtained. Bibliography: 16 titles.</p>