Question

A determinant is chosen at random from the set of all determinants of order 2 with elements 0 and 1 only. The probability that the determinant chosen is positive, is:

• A. $\frac{1}{16}$
• B. $\frac{1}{8}$
• C. $\frac{3}{16}$
• D. $\frac{1}{4}$
• E. $\frac{1}{2}$

Answer 1

Answer: (C)

Since, each entry element of a $2\times 2$ matrix, with elements 0 and 1 only, can be filled in 2 ways $\therefore$ Total $2\times 2$ matrix $=2^4=16$ There are three $2\times 2$ matrices whose determinants are positive i.e., $\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right],\left[\begin{array}{cc}1& 0\\ 1& 1\end{array}\right]$ and $\left[\begin{array}{cc}1& 1\\ 0& 1\end{array}\right]$ . Hence, required probability $=\frac{3}{16}$ .