TIFR ECE 2016 | Question: 13

Question

Suppose $m$ and $n$ are positive integers, $m \neq n$, and $A$ is an $m \times n$ matrix with real entires. Consider the following statements.

1. $\operatorname{rank}\left(A A^{T}\right)=\operatorname{rank}\left(A^{T} A\right)$
2. $\operatorname{det}\left(A^{T} A\right)=\operatorname{det}\left(A A^{T}\right)$
3. $\operatorname{Trace}\left(A A^{T}\right)=\operatorname{Trace}\left(A^{T} A\right)$

Which of the above statements is true for all such $A?$ 

• A. Only (i)
• B. Only (ii)
• C. Only (iii)
• D. (i) and (iii)
• E. None of them