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Bound the Rank of BA Given Rank of AB

Goldman SachsData Scientist · Online Assessment
EasyData ScientistOnline AssessmentStatistics & Math

Last updated

Determine the set of possible values for the rank of the matrix product BA given that A and B are 4x4 real matrices and rank(AB) = 2. This question tests fundamental linear algebra inequalities involving matrix ranks and nullities.

Problem Statement

Let A and B be two real 4 x 4 matrices such that:

rank(AB) = 2

Find all possible values that rank(BA) can take, and provide explicit examples (or structural constructions) justifying the minimum and maximum possible ranks.

Tasks

  1. State the complete set of integers {r_min, ..., r_max} that rank(BA) can achieve under the given constraint.
  2. Use standard rank inequalities (such as Sylvester's rank inequality or properties of null spaces) to establish the theoretical lower and upper bounds on rank(BA).
  3. For each boundary value (minimum and maximum), construct or describe explicit 4 x 4 matrices A and B satisfying rank(AB) = 2 that achieve that rank.

Hints

Hint 1

Recall Sylvester's rank inequality: for n x n matrices, rank(XY) >= rank(X) + rank(Y) - n, and rank(XY) <= min(rank(X), rank(Y)).

Hint 2

Consider whether it is possible for BA to have rank 0 even if AB has rank 2, and what the dimensions of the kernel and image would imply.

Hint 3

Notice that the trace of AB equals the trace of BA, but rank does not generally commute: think about nilpotent matrices or projections where AB is non-zero but BA is zero.

Follow-up questions

  • How does the range of possible values for rank(BA) change if A and B are n x n matrices and rank(AB) = k?
  • If we additionally require that A and B are symmetric, does the minimum possible rank of BA change?
  • Explain how the non-zero eigenvalues of AB compare to those of BA, and whether their algebraic and geometric multiplicities must match.