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JEE Main 202524 JanuaryEvening Shift
Exam
JEE Main
Year
2025
Date
24 January
Shift
Evening Shift
Subject
Mathematics
Chapter
Probability

JEE Main 2025 · 24 January · Evening Shift · Mathematics · Probability

Let A=[a_ i j ] be a square matrix of order 2 with entries either 0...

Let A=[aij]beasquarematrixoforder2withentrieseither0or1.LetA=\left[a_{i j}\right] be a square matrix of order 2 with entries either 0 or 1 . LetEbetheeventthatbe the event thatAisaninvertiblematrix.Thentheprobabilityis an invertible matrix. Then the probability\mathrm{P}(\mathrm{E}) is :

Options

  1. A316\frac{3}{16}
  2. B58\frac{5}{8}
  3. C38\frac{3}{8}
  4. D18\frac{1}{8}

Correct answer

C. 38\frac{3}{8}

Step-by-step solution

A=[ay]2×2\left[a_y\right]_{2 \times 2} and entries are 0 or 1. ∴∣abcd∣=0⇒ad−bc=0\begin{aligned} & \therefore\left|\begin{array}{ll} a & b \\ c & d \end{array}\right|=0 \\ & \Rightarrow a d-b c=0 \end{aligned} Case I: ad=bc=1∴a=b=c=d=1a d=b c=1 \therefore \quad a=b=c=d=1 Case II: ad=bc=0a d=b c=0a=0,d=0b=0,c=0a=0,d=1b=0,c=1a=1,d=0b=1,c=0\begin{array}{ll} a=0, d=0 & b=0, c=0 \\ a=0, d=1 & b=0, c=1 \\ a=1, d=0 & b=1, c=0 \end{array}∴\therefore Total 10 cases when matrix is non invertible Total possible matrix =24=16Requiredprobabilityof=2^4=16Required probability ofinvertible =$\frac{16-10}{16}=\frac{6}{16}=\frac{3}{8}
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