The era of genomic evaluation has brought the need to perform computations involving large, dense matrices. Particular tasks are the computation and inversion of the genomic relationship matrix. This paper investigates the suitability of Graphics Processing Units together with highly optimised software libraries for these computations, using blocked algorithms. It is shown that calculations are readily sped up by parallel processing, using freely available library routines, and that reductions in time by factors of 4 to 5 are achievable even for 'consumer' grade graphics cards.