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Question

Consider the set $A$ of all determinants of order 3 with entries 0 or 1 only. Let $B$ be the subset of $A$ consisting of all determinants with value 1. Let C be the subset of the set of all determinants with value $negative 1.$ Then

$C$ is empty
$B$ has as many elements as $C$
$A = B \cup C$
$B$ has twice as many elements as $C$

The correct answer is: $B$ has as many elements as $C$

Since the value of a determinant charges by minus sign by interchanging any two rows or columns. Therefore, corresponding to every element $increment$ of $B$ there is an element $increment ´$ in $C$ obtained by interchanging two adjacent rows (or columns) in $increment.$ It follows from this that $n left parenthesis B right parenthesis less or equal than n left parenthesis C right parenthesis$
Similarly, we have $n left parenthesis C right parenthesis less or equal than n left parenthesis B right parenthesis$
Hence, $n open parentheses B close parentheses equals n left parenthesis C right parenthesis$

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