TIFR Mathematics 2025 | Part A | Question: 14

Question

Consider the following statements:

1. For any linear map $T: \mathbb{R}^{3} \rightarrow \mathbb{R}^{3}$ with a two-dimensional kernel, the trace of $T$ is an eigenvalue of $T$.
2. For any linear map $T: \mathbb{R}^{3} \rightarrow \mathbb{R}^{3}$ with a two-dimensional kernel, $X^{2}$ divides the characteristic polynomial of $T$.

Which of the following statements is correct?

• A. $\text{(i)}$ and $\text{(ii)}$ are both true.
• B. $\text{(i)}$ and $\text{(ii)}$ are both false.
• C. $\text{(i)}$ is true and $\text{(ii)}$ is false.
• D. $\text{(i)}$ is false and $\text{(ii)}$ is true.

Answer 1

 

3D universe ($\mathbb{R}^3$) = 3 eigenvalues

Eigenvalues be $0, 0, e$.

Let's call this $e =$ $\text{Survivor Eigenvalue}$

Trace $= 0 + 0 + e = e$

So, Trace is an eigenvalue.

Statement (i) is TRUE

Now, $(X - 0)(X - 0)(X - \text{Survivor Eigenvalue})$.

$= X^2(X - \text{Survivor Eigenvalue})$

Hence, Statement (ii) is TRUE

Option A = Correct