Research Scholarships Screening Test Saturday, January 19, 2019 Q. 1.7

Question

Let \( A \) be an \( m \times n \) matrix with real entries. Which of the following statements are true?

     a. \( \operatorname{rank}(A^\top A) \leq \operatorname{rank}(A) \).
     b. \( \operatorname{rank}(A^\top A) = \operatorname{rank}(A) \).
     c. \( \operatorname{rank}(A^\top A) > \operatorname{rank}(A) \).

 

Answer 1

For any real matrix $A$:
\[
\operatorname{rank}(A^\top A) = \operatorname{rank}(A A^\top) = \operatorname{rank}(A) = \operatorname{rank}(A^\top)
\]

Option B = Correct