Verified Solution • 2025–26 Session moderate

QJEEM20Mae3ea: JEE Main 2020 Mathematics - Class 12 JEEM20Mae3ea Solution: Let A be a 2 2 real matrix with entries from {0, 1} and

Let A be a 2 ×\times 2 real matrix with entries from {0, 1} and |A| ≠\ne 0. Consider the following two statements :

(P) If A ≠\ne I2 , then |A| = –1
(Q) If |A| = 1, then tr(A) = 2,

where I2 denotes 2 ×\times 2 identity matrix and tr(A) denotes the sum of the diagonal entries of A. Then :

Multiple Choice Options

  • A (P) is true and (Q) is false
  • B Both (P) and (Q) are false
  • C Both (P) and (Q) are true
  • D (P) is false and (Q) is true

Step-by-Step Verified Solution

Let A = [abcd], where a, b, c, d ∈\in {0, 1}

⇒\Rightarrow |A| = ad – bc

∴\therefore ad = 0 or 1 and bc = 0 or 1

So possible values of |A| are 1, 0 or –1

(P) If A ≠\ne I2 then |A| is either 0 or –1

(Q) If |A| = 1 then ad = 1 and bc = 0

⇒\Rightarrow a = d = 1 ⇒\Rightarrow Tr(A) = 2

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