Let A and B be two n x n matrices over real numbers. Let rank(M) and det(M) denote the rank and determinant of a matrix M, respectively. Consider the following statements.
I. rank(AB) = rank(A) * rank(B)
II. det(AB) = det(A) * det(B)
III. rank(A + B) <= rank(A) + rank(B)
IV. det(A + B) <= det(A) + det(B)
Which of the above statements are TRUE?
A. I and II only B. I and IV only C. II and III only D. III and IV only
GATE 2020 · Engineering Mathematics · Rank of Matrix · medium
Answer: Statements II and III are TRUE. Answer: C. II and III only.
Evaluate Statement I: rank(AB) = rank(A) * rank(B): Counterexample: Let A = B = [[1,0],[0,0]] (rank 1 each). AB = [[1,0],[0,0]] has rank 1. But rank(A)*rank(B) = 1*1 = 1. This case agrees, but try A = [[1,1],[1,1]] (rank 1) and B = [[1,-1],[-1,1]] (rank 1). AB = [[0,0],[0,0]] has rank 0, but 1*1 = 1. So I is FALSE.
Evaluate Statement II: det(AB) = det(A) * det(B): This is the multiplicative property of determinants, a standard theorem in linear algebra. It holds for all n x n square matrices over any field.
Evaluate Statement III: rank(A+B) <= rank(A) + rank(B): This follows because the column space (or row space) of A+B is contained in the span of the column spaces of A and B. The dimension of a sum of subspaces is at most the sum of their dimensions.
Evaluate Statement IV: det(A+B) <= det(A) + det(B): Counterexample: A = B = [[1,0],[0,1]] (det = 1 each). A+B = [[2,0],[0,2]], det(A+B) = 4. But det(A)+det(B) = 1+1 = 2. Here 4 > 2, so IV is FALSE.