From set of all 2 × 2 matrices that can be formed with elements 0 or 1 one matrix is selected at random. Then probability that selected matrix is

$\mathrm{A}\left[\begin{array}{l}-\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}-\\ -\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}-\end{array}\right]$each place can be filled in 2-ways
n(S) 2 x2 x2x 2x 2⁴ 16ways 
A) A square matrix is invertible if its determinants is non zero, it can zero in following ways
But $\left\{\begin{array}{r}\left[\begin{array}{ll}1& 1\\ 0& 0\end{array}\right],\left[\begin{array}{ll}1& 0\\ 0& 0\end{array}\right]\left[\begin{array}{ll}0& 0\\ 1& 0\end{array}\right]\left[\begin{array}{ll}0& 0\\ 0& 1\end{array}\right]\left[\begin{array}{ll}0& 0\\ 1& 1\end{array}\right]\left[\begin{array}{ll}0& 0\\ 1& 1\end{array}\right]\\ \left[\begin{array}{ll}1& 0\\ 1& 0\end{array}\right]\left[\begin{array}{ll}0& 1\\ 0& 1\end{array}\right]\left[\begin{array}{ll}0& 0\\ 0& 0\end{array}\right]\left[\begin{array}{ll}1& 1\\ 1& 1\end{array}\right]\end{array}\right\}$
Number of ways det is zero is 10 $\Rightarrow$ It is not zero 6 ways
$\mathrm{P}\left(\mathrm{E}=\frac{3}{8}\right)=\frac{6}{16}=\frac{3}{8}$
B) $\mathrm{P}\left(\mathrm{E}\right)=\frac{10}{16}=\frac{5}{8}$
C) trace shold be $2\Rightarrow \left[\begin{array}{cc}1& -\\ -& 1\end{array}\right]$remaining 2 place can be followed in 2 × 2 = 4 ways
$\mathrm{P}\left(\mathrm{E}\right)=\frac{4}{16}=\frac{1}{4}$