  Censor-Hillel et al. [PODC'15] recently showed how to efficiently implement
centralized algebraic algorithms for matrix multiplication in the congested
clique model, a model of distributed computing that has received increasing
attention in the past few years. This paper develops further algebraic
techniques for designing algorithms in this model. We present deterministic and
randomized algorithms, in the congested clique model, for efficiently computing
multiple independent instances of matrix products, computing the determinant,
the rank and the inverse of a matrix, and solving systems of linear equations.
As applications of these techniques, we obtain more efficient algorithms for
the computation, again in the congested clique model, of the all-pairs shortest
paths and the diameter in directed and undirected graphs with small weights,
improving over Censor-Hillel et al.'s work. We also obtain algorithms for
several other graph-theoretic problems such as computing the number of edges in
a maximum matching and the Gallai-Edmonds decomposition of a simple graph, and
computing a minimum vertex cover of a bipartite graph.
