Let A and B be 3 × 3 real matrices such that A is symmetric matrix and B is skew-symmetric matrix. Then the system 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 :

Solution

Maths sir · 4 steps

Take the transpose of the coefficient matrix: it comes out skew-symmetric of order 3, so its determinant is 0.

1. Name the matrix. Let C = A²B² - B²A². Since Aᵀ = A, we get (A²)ᵀ = A². Since Bᵀ = -B, we get (B²)ᵀ = (Bᵀ)² = (-B)² = B². So both A² and B² are symmetric.
2. Transpose C. Use (PQ)ᵀ = QᵀPᵀ on each product. This gives Cᵀ = B²A² - A²B² = -C, so C is skew-symmetric.

   det C = det(Cᵀ) = det(-C) = (-1)³ det C = -det C ⇒ det C = 0

3. Homogeneous system. CX = O always has the solution X = O, so it is never inconsistent. With det C = 0, that solution is not the only one, and the system has infinitely many solutions.
4. So. The system has infinitely many solutions, which is option (c).

The idea to keep. A skew-symmetric matrix of odd order always has determinant 0. For a homogeneous system, det = 0 means infinitely many solutions and det ≠ 0 means only X = O.

Answer from the NTA final answer key. Solution by PrepWiser.