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GATE CSE 2020 Q37: Rank and determinant identities for products and sums

Question

Let AA and BB be two n×nn \times n matrices over real numbers. Let rank(M)\text{rank}(M) and det⁡(M)\det(M) denote the rank and determinant of a matrix MM, respectively. Consider the following statements.

I. rank(AB)=rank(A) rank(B)\text{rank}(AB) = \text{rank}(A)\,\text{rank}(B)

II. det⁡(AB)=det⁡(A)det⁡(B)\det(AB) = \det(A)\det(B)

III. rank(A+B)≤rank(A)+rank(B)\text{rank}(A + B) \le \text{rank}(A) + \text{rank}(B)

IV. det⁡(A+B)≤det⁡(A)+det⁡(B)\det(A + B) \le \det(A) + \det(B)

Which of the above statements are TRUE?

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Answer: C (official GATE 2020 final answer key)

Concept used: The determinant is multiplicative and rank is subadditive; neither has the other property.

First step: Test I and IV with A=B=I2A = B = I_2: rank(AB)=2≠4\text{rank}(AB) = 2 \ne 4 and det⁡(2I2)=4>2\det(2I_2) = 4 > 2.

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Source: official GATE 2020 CS question paper and final answer key, organised by IIT Delhi. Question paper · Answer key

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