Let A be a 2 × 2 real matrix with entries from 0 ,   1 and A ≠ 0 . Consider the following…

Let A be a 2×2 real matrix with entries from 0, 1 and A≠0. Consider the following two statements;

P If A≠l2, then A=-1

Q If A=1, then trA=2

Where l2 denotes 2×2 identity matrix and trA denotes the sum of the diagonal entries of A. Then

  1. P is false and Q is true
  2. Both P and Q are false
  3. P is true and Q is false
  4. Both P and Q are true

Solution

Let A=abcd  a,b,c,d∈{0,1}

A=ad-bc≠0

⇒ ad=1,bc=0  or  ad=0,bc=1

P

If A≠I2⇒ad≠1⇒ad=0,bc=1⇒|A|=-1

P is true.

Q 

If A=1⇒ad=1,bc=0⇒ tr(A)=2

Q is true.

Asked in: JEE Main 2020 (02 Sep Shift 1)

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