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Two matrices $A$ and $B$ are called similar if there exists an invertible matrix $X$ such that $A=X^{-1} B X$. Let $A$ and $B$ be two similar matrices. Consider the following statements:

  1. $\operatorname{det}(x I-A)=\operatorname{det}(x I-B)$ for any scalar $x$
  2. The eigenvalues of $A$ and $B$ are identical
  3. $\left[\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right]$ and $\left[\begin{array}{ll}0 & 1 \\ 1 & 0\end{array}\right]$ are similar

Which of the following is $\text{TRUE?}$

  1. Only statement $1$ is correct
  2. Only statement $2$ is correct
  3. Only statements $1$ and $2$ are correct
  4. All Statements $1, 2$ and $3$ are correct
  5. None of the above
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