# Problem in matrices

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|

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

• Switching the first two rows of any matrix with determinant $$1$$ produces a matrix with determinant $$-1$$