Question

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 ______.

विकल्प

• `133/10^4`

• `18/10^3`

• `19/10^3`

• `271/10^4`

Answer 1

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 `underlinebb(19/10^3)`.

Explanation:

Let M be a 2 × 2 matrix such that M = `[(m, n),(o, p)]` and For M to be a singular matrix, |M| = 0

⇒ mp – on = 0

Case 1: All four elements are equal

m = n = o = p

⇒ mp – on = 0

So, number of matrices possible = 10

Case 2: When two prime numbers are used

⇒ Either m = n and o = p or m = o and n = p

So, number of matrices possible = ¹⁰C₂ × 2! × 2!

⇒ `(10 xx 9)/2 xx 2 xx 2`

= 180

So, number of matrices possible = 10 + 180 = 190

And total number of matrices that can be formed = 10 × 10 × 10 × 10 = 10⁴

So, required probability = `190/10^4` = `19/10^3`