SummaryThe generation matrix is derived for the effects of sib‐mating on two linked loci (a, b) and (A, B), when selection against homozygosity acts at the (A, B) locus so that only fractions x of A A and BB survive, but the (a, b) locus is not directly affected by selection. The matrix is of order 19 × 19. Using an electronic digital computer, the rates of inbreeding progress at the (a, b) locus have been calculated for various survival values x and recombination frequencies y, under the following types of selection:Case I–selection acts within lines only.Case II–selection acts equally within and between lines.The results quoted include, for various (x, y), the final rate of change in heterozygosity λ (Table 4), the relative speed of inbreeding after 10,20 and a large number of generations (Table 5), and the number of heterozygotes per 1000 remaining after 5, 10 and 20 generations (Table 6).Graphs illustrate the effect of changing x, when y is constant, on the relative speed of inbreeding after 20 generations (Fig. 1) and after a large number of generations when this speed has become constant (Fig. 2). At generation 20 the speed declines fairly steadily with x in case I, reaching a minimum with x between 0–1 and 0‐0, after which it rises slightly; but in case II the speed falls rapidly to a minimum with x about 0–6, and then rises slowly to the same value as in case I when x= 0.The final speed of inbreeding for constant y does not generally fall below 100 % in case I until x is well below unity, and then falls fairly rapidly to a minimum with x close to zero. The case II lines drop below 100 % with x a little below unity, fall very rapidly to a minimum at x= 0–7633, and then rise slowly to join the case I lines at x= 0. Thus, under case II selection, inbreeding progress at a linked locus is reduced most when the selection pressure is only of moderate strength.The 19 × 19 generation matrix contains 4 submatrices along the leading diagonal with zeros below each, so that their roots are the latent roots of the complete matrix. These matrices are A1 (both loci fixed), A2 (locus (a, b) fixed), A3 (locus (A, B) fixed) and A4 (neither locus fixed). It is shown that the final rate of change in heterozygosity (A) is the largest of the latent roots of A3 and A4 in case I, and is the ratio of the largest latent root of A3 and A4 to the largest latent root of A1 and A2 in case II, for any particular (x, y).