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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 $\text{TRUE}?$

  1. Only $\text{(i)}$
  2. Only $\text{(ii)}$
  3. Only $\text{(iii)}$
  4. Only $\text{(iv)}$
  5. All of $\text{(i) – (iv)}$
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