Verified Solution • 2025–26 Session moderate

QJEEM22Ma1fc6: JEE Main 2022 Mathematics - Class 12 JEEM22Ma1fc6 Solution: Let A and B be two non-zero real matrices such that AB is a…

Let A and B be two 3×33 \times 3 non-zero real matrices such that AB is a zero matrix. Then

Multiple Choice Options

  • A the system of linear equations AX=0A X=0 has a unique solution
  • B the system of linear equations AX=0A X=0 has infinitely many solutions
  • C B is an invertible matrix
  • D adj⁡(A)\operatorname{adj}(\mathrm{A}) is an invertible matrix

Step-by-Step Verified Solution

AB is zero matrix

⇒∣A∣=∣B∣=0\Rightarrow |A| = |B| = 0

So neither A nor B is invertible

If ∣A∣=0|A| = 0

⇒∣adj A∣=0\Rightarrow |\mathrm{adj}\,A| = 0 so adj A\mathrm{adj}\,A

AX=0AX = 0 is homogeneous system and ∣A∣=0|A| = 0

So, it is having infinitely many solutions

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