GATE CSE 2020 | Question: 27

Question

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

1. $\text{rank}(AB) = \text{rank }(A) \text{rank }(B)$
2. $\text{det}(AB) = \text{det}(A) \text{det}(B)$
3. $\text{rank}(A + B) \leq \text{rank }(A) + \text{rank }(B)$
4. $\text{det}(A + B) \leq \text{det}(A) + \text{det} (B)$

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

Comments

https://math.stackexchange.com/questions/3006694/what-does-it-mean-geometrically-to-add-two-matrices#:~:text=1%20Answer&text=Linearity%20works%20both%20ways.,B%2C%20separately%2C%20added%20together.

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for statement 1 proof

https://ocw.mit.edu/courses/6-241j-dynamic-systems-and-control-spring-2011/de27b7d21756e72fba5395802d5cb962_MIT6_241JS11_assn02_sol.pdf

see exercise 1.4 (b)

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C

Answer 1

$\textsf{Rank}(AB)=\min(\textsf{Rank}(A),\textsf{Rank}(B))$

$\textsf{Det}(AB)=\textsf{Det}(A) \times \textsf{Det}(B)$

$\textsf{Rank}(A+B) \leq \textsf{Rank}(A)+\textsf{Rank}(B).$ Because addition of two matrices can never result in increase in the number of independent columns and rows in the matrix.

Answer: C

Comments

Thanks sir

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What about the explanation for fourth one !  It seems that 4^th statement is actually very hard to answer even by researchers, quite possibly there is no relationship 

 

please go through this link : https://sites.lafayette.edu/thompsmc/files/2015/08/Section_2_31.pdf

 

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@s_dr_13 from GATE exam  point of view you can simply take an example  and simply prove that fourth one is not valid.

Answer 2

Statement II and III are correct statements directly based on the properties of matrices.

C. II and III only.

Answer 3

(c.)  det(AB) = det(A)*det(B) 

      RANK(A+B)<= RANK(A)+RANK(B)