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Let $S =\{A=[a_{ij}]_{n\times n}:a_{ij}\in\{0,1,2,3,4,5\}\}$, B and C are subset of S s.t. $B=\{A\in S:|A|=1\}$ and $C=\{A\in S:|A|=-1\}$. Then for n>=2 which is true..

1)B and C are finite sets.
2)B and C are infinite sets.
3)|B|=|C| if n>=2
4)|B| not equal to |C|

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  • $\begingroup$ What are your thoughts on this problem? What have you tried so far? $\endgroup$ – Omnomnomnom Sep 11 at 16:36
  • $\begingroup$ Also, does $|A|$ refer to the determinant of $A$ in this context? $\endgroup$ – Omnomnomnom Sep 11 at 16:37
  • $\begingroup$ Yes..|A|= det A, but since B, C are sets then |B|,|C| refers to number of elements in the sets B and C. $\endgroup$ – Abhinandan Saha Sep 11 at 16:46
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Hint:

  • Any subset of a finite set is finite
  • Switching the first two rows of any matrix with determinant $1$ produces a matrix with determinant $-1$
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