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{matrix} 1 & 0 \\ 0 & 1 \\ \end{matrix} \right],\left[ \begin{matrix} 1 & 0 \\ 1 & 1 \\ \end{matrix} \right] $ and $ \left[ \begin{matrix} 1 & 1 \\ 0 & 1 \\ \end{matrix} \right] $ . Hence, required probability $ =\frac{3}{16} $ .