Question

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

• A. $\frac{3}{16}$
• B. $\frac{6}{16}$
• C. $\frac{10}{16}$
• D. $\frac{13}{16}$

Answer 1

The correct option is D $\frac{13}{16}$

A determinant of a matrix of order $2$ is chosen with elements $0$ or $1$.
Total no. of combinations can be formed for four places in $2\times 2\times 2\times 2=16$ ways
If $\left[\begin{array}{c}ab\\ cd\end{array}\right]$ is a matrix, the determinant of matrix $=ad-bc$
For determinant to be negative, $ad-bc<0$

Now, $(a,b,c,d)=(0,1,1,0),(1,1,1,0),(0,1,1,1)$ for $ad-bc<0$
$\therefore$ Probability for determinant to be negative $=\frac{3}{16}$
$\therefore$ Probability for determinant to be of non-negative value $=1-\frac{3}{16}=\frac{13}{16}$