TIFR ECE 2013 | Question: 11

Question

Two matrices $A$ and $B$ are called similar if there exists another matrix $S$ such that $S^{-1} A S=B$. Consider the statements:

1. If $A$ and $B$ are similar then they have identical rank.
2. If $A$ and $B$ are similar then they have identical trace.
3. $A=\left[\begin{array}{ll}1 & 0 \\ 0 & 0\end{array}\right]$ and $B=\left[\begin{array}{ll}1 & 0 \\ 1 & 0\end{array}\right]$ are similar.

Which of the following is TRUE.

• A. Only $\text{I}$.
• B. Only $\text{II}$.
• C. Only $\text{III}$.
• D. Both $\text{I}$ and $\text{II}$ but not $\text{III}$.
• E. All of $\text{I}, \text{II}$ and $\text{III}$.