If A is symmetric matrix and B is skew symmetric matrix of order `3xx3`, then consider `(A^2B^2-B^2A^2)X=0`, where X is a matrix of unknown variable of `3xx1` and O is a null matrix of `3xx1` , then system of linear equation has

Options

A. No Solution
B. infinity Solution
C. Unique Solution
D. None of these

Answer

To solve the problem, we need to analyze the given equation \( (A^2B^2 - B^2A^2)X = 0 \), where \( A \) is a symmetric matrix and \( B \) is a skew-symmetric matrix. We will follow these steps:

### Step 1: Understand the properties of symmetric and skew-symmetric matrices
- A symmetric matrix \( A \) satisfies \( A^T = A \).
- A skew-symmetric matrix \( B \) satisfies \( B^T = -B \).

### Step 2: Define the matrix \( P \)
Let \( P = A^2B^2 - B^2A^2 \). We need to analyze the properties of \( P \).

### Step 3: Take the transpose of \( P \)
To find the properties of \( P \), we will take its transpose:
\[ P^T = (A^2B^2 - B^2A^2)^T = (B^2A^2)^T - (A^2B^2)^T \]
Using the property of transposes, we have:
\[ (B^2A^2)^T = A^2B^2 \quad \text{(since \( B^2 \) is symmetric)} \]
\[ (A^2B^2)^T = B^2A^2 \quad \text{(since \( A^2 \) is symmetric)} \]
Thus, we can rewrite \( P^T \):
\[ P^T = A^2B^2 - B^2A^2 = P \]

### Step 4: Show that \( P \) is skew-symmetric
From the previous step, we have \( P^T = -P \). This implies that \( P \) is skew-symmetric.

### Step 5: Determine the determinant of \( P \)
For a skew-symmetric matrix of odd order (in this case, \( 3 \times 3 \)), the determinant is always zero. Therefore:
\[ \text{det}(P) = 0 \]

### Step 6: Analyze the system of equations
The equation \( PX = 0 \) represents a homogeneous system of linear equations. Since the determinant of the coefficient matrix \( P \) is zero, this indicates that the system has either no solutions or infinitely many solutions.

### Step 7: Conclusion
In the case of a homogeneous system, if the determinant of the coefficient matrix is zero, it implies that there are infinitely many solutions. Therefore, the correct answer is:

**Infinite solutions.**