Question

Let A and B be 3 × 3 real matrices such that A is symmetric matrix and B is skew-symmetric matrix. Then the systems of linear equations (A²B² – B²A²)X = O, where X is a 3 × 1 column matrix of unknown variables and O is a 3 × 1 null matrix, has ______.

विकल्प

• no solution

• exactly two solutions

• infinitely many solutions

• a unique solution

Answer 1

Let A and B be 3 × 3 real matrices such that A is symmetric matrix and B is skew-symmetric matrix. Then the systems of linear equations (A²B² – B²A²)X = O, where X is a 3 × 1 column matrix of unknown variables and O is a 3 × 1 null matrix, has infinitely many solutions.

Explanation:

Let A^T = A and B^T = – B

C = A²B² – B²A²

C^T = (A²B²)^T – (B²A²)^T   ...[∵ (AB)^T = B^TA^T] 

= (B²)^T(A²)^T –  (A²)^T(B²)^T 

= B²A² – A²B² 

C^T = – C

`\implies` C is skew-symmetric.

So, det (C) = 0.

So, systems have infinite solutions.