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

Let A and B be two 3×3 non-zero real matrices such that AB is a zero matrix. Then
  1. The system of linear equations AX=0 has a unique solution
  2. The system of linear equations AX=0 has infinitely many solutions
  3. B is an invertible matrix
  4. adjA is an invertible matrix

Solution

Given,

AB=0⇒AB=0

If A≠0, B=0 (not possible)

If B≠0, A=0  (not possible)

Hence A=B=0

So, if A=B=0 then A & B are non-invertible matrix,

So,  AX=0 has infinitely many solutions

Asked in: JEE Main 2022 (29 Jul Shift 1)

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