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CMI-2020-DataScience-A: 8

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Let $A=((a_{ij}))$ be a $7\times7$ matrix with $a_{i,i+1}=1$ for $1\leq i \leq 6$, $a_{7,1}=1$ and all the other elements of the matrix are zero. Which of the following statements are true?

  1. $|A|=1$
  2. $\text{trace(A)}=0$
  3. $A^{-1}=A$
  4. $A^7 =I$, where $I$ is the identity matrix
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