Question

A $2\times 2$ matrix is formed with entries from the set $\{0,1\}$ . The probability that it is singular is $8k$ . Find $k$

Answer 1

Answer: 5

In $2\times 2$ matrices total positions are $4$ in each position there are $2$ choices.
$\therefore n(S)=2^4=16$
Matrices which are not singular are
$\left[\begin{array}{cc}1& 1\\ 0& 1\end{array}\right],\left[\begin{array}{cc}0& 1\\ 1& 1\end{array}\right],\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right],\left[\begin{array}{cc}0& 1\\ 1& 0\end{array}\right]$
$\left[\begin{array}{cc}1& 0\\ 1& 1\end{array}\right],\left[\begin{array}{cc}1& 1\\ 1& 0\end{array}\right]$
$\therefore$ Total matrices which are singular
$=16-6=10$
$\therefore$ Required probability
$=\frac{10}{16}=\frac{5}{8}=0.63$
$\Rightarrow 8k=5$