GATE CSE 2022 | Question: 10

Question

Consider the following two statements with respect to the matrices $\textit{A}_{m \times n}, \textit{B}_{n \times m}, \textit{C}_{n \times n}$ and $ \textit{D}_{n \times n}.$

Statement $1: tr \text{(AB)} = tr \text{(BA)}$

Statement $2: tr \text{(CD)} = tr \text{(DC)}$

where $tr()$ represents the trace of a matrix. Which one of the following holds?

• A. Statement $1$ is correct and Statement $2$ is wrong.
• B. Statement $1$ is wrong and Statement $2$ is correct.
• C. Both Statement $1$ and Statement $2$ are correct.
• D. Both Statement $1$ and Statement $2$ are wrong.

Comments

Link to such theorems. Might be helpful.

Trace (linear algebra) From Wikipedia, the free encyclopedia

Answer 1

Answer C

Trace of a matrix and sum of eigen values of matrix are same.

$AB$ and $BA$ has same nonzero eigen values (Proof below) hence their sum of eigen values i.e.  trace should also be same.

 

Proof-

Comments

Sir, Did you explain this in GO class lectures?

---

@s_dr_13 yes, above screenshots are from lectures.

---

@Sachin Mittal 1 Sir has made answering such qs damn easy by relating them to Eigen Values :)

---

great concept

Answer 2

Statement -1 is evident from the definition of matrix product 

 

Statement-2 follows from statement-1, just that now both are square matrices

 

So, both are TRUE 

option C