Question

A determinant is chosen at random from the set of all determinant of order 2 with elements 0 or 1 only. Find the probability that the determinant chosen is nonzero.

• A.
• B.
• C.
• D.

Answer 1

The correct option is C

A determinant of order 2 is of the form
$\Delta =\left|\begin{array}{cc}a& b\\ c& d\end{array}\right|$
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, $\Delta \ne 0$ if and only 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, the required probability is $\frac{6}{16}=\frac{3}{8}$