Question

A determinant is chosen at random from the set of all determinants of order $2$ with elements $0$ or $1$ only. The probability that the determinant is non-zero is

• A. $\frac{3}{16}$
• B. $\frac{3}{8}$
• C. $\frac{5}{8}$
• D. $\frac{7}{8}$

Answer 1

Answer: (B)

$\frac{3}{8}$

$P$ (determinant is non zero) = 1-$P$(determinant is zero)
$\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]$
total no. of elements in determinant of order $2=4$
And each element can have value either $0$ or $1$
Hence, total no. of determinants possible = $16$
Determinant is zero only when $ad=bc$
$Case1$: $a=b=c=d=1$ , determinants = $1$
$Case2$: either of a, d and b,c is zero, no. of determinants possible $=4$
$case3$: three of a,b,c,d are zero, no. of determinants possible $=4$
$case4$ : a=b=c=d = 0 , determinants $=1$
$P$ = 1 - $10/16=3/8$