Consider a determinant D =|a b c d | where a , b , c , d ∈ 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 (m/n) equals

Question

Consider a determinant D =|a b c d | where a , b , c , d ∈ 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 (m/n) equals (A) (3/16) (B) (4/16) (C) (6/16) (D) (8/16)

Tardigrade Admin · Accepted Answer

D=a d-b c ; n=24=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 ⇒ 3 text cases b =0 ; c =0 Case-II: b c=1 ad =0 i.e. a=1 ; d=0 a=0 ; d=1 ⇒ 3 text cases a=0, d=0 Hence total determinant with non zero value =6 ∴ (m/n)=(6/16)

Q. Consider a determinant $D = ∣ ∣ a c b d ∣ ∣$ 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

$\frac{3}{16}$

$\frac{4}{16}$

$\frac{6}{16}$

$\frac{8}{16}$

Q. Consider a determinant D=∣∣​ac​bd​∣∣​ where a,b,c,d∈{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 nm​ equals

163​

164​

166​

168​

D=ad−bc;n=24=16;D can be non zero if Case-I: ad=1 and bc=0 i.e. b=1;c=0 b=0;c=1⇒3 cases b=0;c=0 Case-II : bc=1& ad =0 i.e. a=1;d=0 a=0;d=1⇒3 cases a=0,d=0 Hence total determinant with non zero value =6 ∴nm​=166​

Q. Consider a determinant $D = ∣ ∣ a c b d ∣ ∣$ 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

$\frac{3}{16}$

$\frac{4}{16}$

$\frac{6}{16}$

$\frac{8}{16}$