GATE CS 2021 Set 1 — Question 28

Let \(A\) and \(B\) be two \(n \times n\) matrices over real numbers. Let \(\operatorname{rank}(M)\) and \(\det(M)\) denote the rank and determinant of a matrix \(M\), respectively. Consider the following statements.

I. \(\operatorname{rank}(AB) = \operatorname{rank}(A)\,\operatorname{rank}(B)\)
II. \(\det(AB) = \det(A)\,\det(B)\)
III. \(\operatorname{rank}(A+B) \le \operatorname{rank}(A) + \operatorname{rank}(B)\)
IV. \(\det(A+B) \le \det(A) + \det(B)\)

Which of the above statements are TRUE?