Question

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

Answer 1

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