Question

Let A be the set of all determinants of order 3 with entries 0 or 1 only. B is the subset of A consisting of all determinants with value 1 and C is the subset of A consisting of all determinants with value -1. If $n(B)$ and $n(C)$ denote the number of elements in B and C respectively, then

• A. C = $\phi$
• B. $n(B) = n(C)$
• C. $A = B \cup C$
• D. $n(B) = 2n(C)$

Answer 1

Answer: (B)

C cannot be the empty set because, for instance,
$-1=\begin{vmatrix} 0& 1& 0 \\[0.3em] 1 & 0 & 0 \\[0.3em] 0 & 0 & 1 \end{vmatrix} \,\in \: C$. We also have
$\begin{vmatrix} 1& 0& 1 \\[0.3em] 1 & 1 & 0 \\[0.3em] 0 & 1 & 1 \end{vmatrix} \,=2$
So A $\neq$ B U C. In general, the determinant
$\begin{vmatrix} a_{11}& a_{12}& a_{13} \\[0.3em] a_{21} & a_{22} & a_{23}\\[0.3em] a_{31} & a_{32} & a_{33} \end{vmatrix} $
$= a_{11}a_{22}a_{33}+a_{12}a_{23}a_{31}+a_{13}a_{21}a_{32}$$-a_{11}a_{23}a_{32}-a_{12}a_{21}a_{33}-a_{13}a_{31}a_{22}$,
with the being 0 or 1, equals 1 only if $a_{11}a_{22}a_{33}$= 1 and the remaining terms are zero ; if $a_{12}a_{22}a_{31}$
= 1 and the remaining terms are zero ; or if $a_{13}a_{21}a_{32}$ = 1 and the remaining terms are zero. Since there are three similar relations for determinants that equal - 1, we must have n(B) = n(C).
$\therefore $ $n(B) = n(C)$ holds.