Some matrix problems over (finite) fields

MM(n): C = AB, where A,B \in F^{n\times n}
Tinv(n): Invert a nonsingular (upper or lower) triangular n by n matrix
inv(n): Invert a nonsingular matrix
LU(n):  Compute a PLE or PLUQ decomposition of A.  Here P,Q are permutations, 
        L is unit lower triangular, E is in row echelon form, 
        U is upper trapezoidal (echelon with first row entries on the diagonal).

DetEqZero(n): Determine if det(A) is zero.
    Nonsingular matrices
Det(n): Determinant of A
SolveNonsingular(n): Solve Ax = b, A n\times n, nonsingular
    Singular matrices
Rank(n): Compute rank of A
SolveArb(n): Compute an arbitrary solution x to consistent system Ax = b.
NullspaceRandom(n): Compute a uniformly random sample of the (right) nullspace.
NullspaceBasis(n): Compute N \in F^{n\times r} of rank r such that AN = 0.

                 Linear time Reductions
Matrix problem P(n^2) -> matrix problem Q(n^2) means there is an algorithm 
for P which calls Q on data of size O(n^2) at most O(1) times and takes 
O(n^2) additional field operations to set up the data for calls to Q, 
handle the results of calls to Q, and do any and all additional work.

Theorem: These problems have linear time reductions to each other: 
   MM, Tinv, inv, LU
Proof Sketch: show reductions MM -> Tinv -> inv -> LU + Tinv -> MM + Tinv -> MM
Theorem: Rank, Det -> LU 
Theorem: SolveNonsing, SolveArb, NullspaceRandom -> LU + Tinv
Theorem: SolveArb <-> NullspaceRandom (Las Vegas)

FIBBs are matrices on which Rank, Det, Solve's, Nullspace's are fast.
Permutations, triangular matrices, and products thereof are FIBBs, subject to a rank condition: A product of FIBBs is a FIBB if at most one factor is singular.

Certifications:
AB <> C?