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 = $\varphi$
• 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,
$-1=\left|\begin{array}{ccc}0& 1& 0\\ 1& 0& 0\\ 0& 0& 1\end{array}\right|\in C$. We also have
$\left|\begin{array}{ccc}1& 0& 1\\ 1& 1& 0\\ 0& 1& 1\end{array}\right|=2$
So A $\ne$ B U C. In general, the determinant
$\left|\begin{array}{ccc}{a}_{11}& a_{12}& a_{13}\\ a_{21}& a_{22}& a_{23}\\ a_{31}& a_{32}& a_{33}\end{array}\right|$
$=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.