The probability that a randomly chosen 2 × 2 matrix with all the...

Let $\mathrm{A}=$\left[$$\begin{array}{ll}\mathrm{a} & \mathrm{b} \\ \mathrm{c} & \mathrm{d}\end{array}$$\right]$where$\mathrm{a}, \mathrm{b}, \mathrm{c}, \mathrm{d}$are prime.<br>Total number of matrices A formed$=10^4$<br>Let$\mathrm{S}$be a matrix from set of matrices$\mathrm{A}$such that$\mathrm{S}$is singular.<br>For singular matrix$|\mathrm{S}|=\mathrm{ad}-\mathrm{bc}=0 \Rightarrow \mathrm{ad}=\mathrm{bc}$<br>Number of singular matrices = All entries are same$+$only two prime number are used in matrix<br>then number of matrices with two prime numbers$=(10 \times 9) \times 2 !=180$<br>then number of matrices with one prime number$=10 \times 1=10$<br>So required probability$=\frac{10 \times 9 \times 2 !+10}{10^4}=\frac{19}{1000}=0.019