The probability that a randomly chosen 2 × 2 matrix with all the entries from the set of first 10…

The probability that a randomly chosen 2×2 matrix with all the entries from the set of first 10 primes, is singular, is equal to
  1. 133104
  2. 19103
  3. 18103
  4. 271104

Solution

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.
Total number of matrices A formed $=10^4$
Let $\mathrm{S}$ be a matrix from set of matrices $\mathrm{A}$ such that $\mathrm{S}$ is singular.
For singular matrix $|\mathrm{S}|=\mathrm{ad}-\mathrm{bc}=0 \Rightarrow \mathrm{ad}=\mathrm{bc}$
Number of singular matrices $=$ All entries are same $+$ only two prime number are used in matrix
then number of matrices with two prime numbers $=(10 \times 9) \times 2 !=180$
then number of matrices with one prime number $=10 \times 1=10$
So required probability $=\frac{10 \times 9 \times 2 !+10}{10^4}=\frac{19}{1000}=0.019$

Asked in: JEE Main 2022 (29 Jun Shift 1)

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