IIT JEE 1982 Maths - 'Adapted' - FIB to Single-Digit Integer Q2

Algebra Level 3

A determinant is chosen at random from the set of all determinants of order 2 with elements 0 or 1 only. The probability that the value of the determinant chosen is positive is p / q p/q , where p p , q q are coprime positive integers. Find the value of q − 3 p q-3p .

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The answer is 7.

Shubhamkar Ayare by Shubhamkar Ayare , 22, India

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1 solution

Tom Engelsman
Feb 2, 2017

There are 16 such 2x2 matrices that have only 0 or 1 for entries:

2 with all 0's or all 1's; 4 with three 0's + one 1; 4 with three 1's + one 0; 6 with two 0's + two 1's.

Of these, only 3 have a positive determinant:

∣ 1 0 0 1 ∣ ; ∣ 1 0 1 1 ∣ ; ∣ 1 1 0 1 ∣ \begin{vmatrix} 1&0\\0&1 \end{vmatrix}; \begin{vmatrix} 1&0\\1&1 \end{vmatrix}; \begin{vmatrix} 1&1\\0&1 \end{vmatrix}

The required probability is p q = 3 16 ⇒ q − 3 p = 16 − 3 ( 3 ) = 7 . \frac{p}{q} = \frac{3}{16} \Rightarrow q - 3p = 16 -3(3) = \boxed{7}.

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