Question

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

• A. $\frac{3}{16}$
• B. $\frac{3}{8}$
• C. $\frac{1}{4}$
• D. None of these

Answer 1

The correct option is A $\frac{3}{16}$

A determinant of order $2$ is of the form $△=\left|\begin{array}{cc}a& b\\ c& d\end{array}\right|$
It is equal to $ad-bc.$ The total number of ways of choosing $a,b,c$ and $d$ is $2\times 2\times 2\times 2=16.$
Now $△\ne 0$ if either $ad=1,bc=0$ or $ad=0,bc=1.$ But $ad=1,bc=0$ if $a=d=1$ and one of $b,c$ is zero.
$\therefore ad=1,bc=0$ in three cases,
similarly $ad=0,bc=1$ in three cases.
$\therefore$ required probability $=\frac{6}{16}=\frac{3}{8}$