Question

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 :

• A. no solution
• B. a unique solution
• C. exactly two solutions
• D. infinitely many solutions

Hint:

Determinant of any skew-symmetric matrix of odd order ($3\times3, 5\times5$, etc.) is always zero.

Answer 1

The Correct Option is D

Step 1: Given $A^T = A$ and $B^T = -B$.
Step 2: Let $C = A^2B^2 - B^2A^2$. We check if $C$ is skew-symmetric.
Step 3: $C^T = (A^2B^2)^T - (B^2A^2)^T = (B^T)^2(A^T)^2 - (A^T)^2(B^T)^2$.
Step 4: $C^T = (-B)^2(A)^2 - (A)^2(-B)^2 = B^2A^2 - A^2B^2 = -C$.
Step 5: $C$ is a $3 \times 3$ skew-symmetric matrix. The determinant of an odd-order skew-symmetric matrix is always 0.
Step 6: Since $|C| = 0$, the homogeneous system $CX = O$ has non-trivial solutions, meaning infinitely many solutions.