# GATE | GATE CS 2020 | Question 37

Let A and B be two n×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

Explanation: I. False. According to properties of rank of matrices:

`if P(A) = m and P(B)=n then P(AB) ≤ min(m,n) `

However, if A and B are square matrices of order n then

`P(AB) = P(A) + P(B) – n `

II. True. That is, the determinant of the product is equal to the product of the determinants.
III. True. Because, resultant matrix of sum of two non-singular matrix can be singular matrix.
IV. False. Because, resultant matrix of sum of two singular matrix can be non-singular matrix.

Option (C) is true.

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