Question

Consider a determinant $D =\begin{vmatrix}a & b \\ c & d \end{vmatrix}$ where $a , b , c , d \in\{0,1\}$. If $n$ denote the total number of determinants whose entries are 0 or 1 only and $m$ denote the number of determinants with non zero value, then the ratio $\frac{m}{n}$ equals

• A. $\frac{3}{16}$
• B. $\frac{4}{16}$
• C. $\frac{6}{16}$
• D. $\frac{8}{16}$

Answer 1

Answer: (C)

$ D=a d-b c ; n=2^4=16 ; D$ can be non zero if
Case-I: $a d=1$ and $b c=0$ i.e.
$ b =1 ; c =0 $
$b =0 ; c =1 \Rightarrow 3 \text { cases }$
$b =0 ; c =0$
Case-II : $b c=1 \&$ ad $=0$ i.e.
$a=1 ; d=0 $
$a=0 ; d=1 \Rightarrow 3 \text { cases } $
$a=0, d=0$
Hence total determinant with non zero value $=6$
$\therefore \frac{m}{n}=\frac{6}{16}$