Singular invariant hyperfunctions on the space of n × n complex and quaternion matrices are discussed in this paper. Following a parallel method employed in the author's paper on invariant hyperfunctions on the symmetric matrix spaces, we give an algorithm to determine the orders of poles of the complex power of the determinant function and to determine exactly the support of singular invariant hyperfunctions, i.e., invariant hyperfunctions whose supports are contained in the set of points of rank strictly less than n, obtained as negative-order-coefficients of the Laurent expansions of the complex powers.