Let A and B be 3 × 3 real matrices such that A is a symmetric matrix and B is a skew-symmetric matrix.…

Let A and B be 3×3 real matrices such that A is a symmetric matrix and B is a skew-symmetric matrix. Then the system of linear equations A2 B2-B2 A2X=O, where X is a 3×1 column matrix of unknown variables and O is a 3×1 null matrix, has
  1. exactly two solutions
  2. infinitely many solutions
  3. a unique solution
  4. no solution

Solution

Let AT=A and BT=-B

C=A2B2-B2A2

CT=A2 B2T-B2 A2T

=B2TA2T-A2TB2T

=B2 A2-A2 B2

CT=-C

C is skew symmetric.

So det C=0

so system have infinite solutions.

Asked in: JEE Main 2021 (24 Feb Shift 2)

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