TIFR ECE 2018 | Question: 8

Question

Let $A$ be an $n \times n$ real matrix for which two distinct non-zero $n$-dimensional real column vectors $v_{1}, v_{2}$ satisfy the relation $A v_{1}=A v_{2}$. Consider the following statements.

1. At least one eigenvalue of $A$ is zero.
2. $A$ is not full rank.
3. Columns of $A$ are not linearly independent.
4. $\operatorname{det}(A)=0$.

Which of the above statements is/are $\text{TRUE}?$

• A. Only $\text{(i)}$
• B. Only $\text{(ii)}$
• C. Only $\text{(iii)}$
• D. Only $\text{(iv)}$
• E. All of $\text{(i) – (iv)}$