If matrix `A=[a_(ij)]_(3xx3)`, matrix `B=[b_(ij)]_(3xx3)`, where `a_(ij)+a_(ji)=0` and `b_(ij)-b_(ji)=0 AA i`, `j`, then `A^(4)*B^(3)` is

Options

A. Singular
B. Zero matrix
C. Symmetric
D. Skew-Symmetric matrix

Explanation

To solve the problem, we need to analyze the properties of the matrices \( A \) and \( B \) based on the conditions provided, and then determine the nature of the product \( A^4 B^3 \).

### Step-by-Step Solution:

1. **Identify the properties of matrix \( A \)**:
- We are given that \( a_{ij} + a_{ji} = 0 \) for all \( i, j \).
- This condition indicates that matrix \( A \) is skew-symmetric. A skew-symmetric matrix has the property that \( A^T = -A \).

2. **Determine the order of matrix \( A \)**:
- The matrix \( A \) is a \( 3 \times 3 \) matrix (order 3).
- For skew-symmetric matrices of odd order, it is known that the determinant is zero.

3. **Calculate the determinant of matrix \( A \)**:
- Since \( A \) is skew-symmetric and of odd order, we have:
\[ \text{det}(A) = 0 \]

4. **Identify the properties of matrix \( B \)**:
- We are given that \( b_{ij} - b_{ji} = 0 \) for all \( i, j \).
- This condition indicates that matrix \( B \) is symmetric. A symmetric matrix has the property that \( B^T = B \).

5. **Calculate the determinant of matrix \( B \)**:
- The determinant of matrix \( B \) can be any value (not necessarily zero), but we will denote it as \( \text{det}(B) \).

6. **Calculate the determinant of \( A^4 B^3 \)**:
- We can use the property of determinants that states:
\[ \text{det}(A^m B^n) = \text{det}(A)^m \cdot \text{det}(B)^n \]
- Therefore, we have:
\[ \text{det}(A^4 B^3) = \text{det}(A^4) \cdot \text{det}(B^3) = (\text{det}(A))^4 \cdot (\text{det}(B))^3 \]
- Substituting the values we found:
\[ \text{det}(A^4 B^3) = (0)^4 \cdot (\text{det}(B))^3 = 0 \]

7. **Conclusion about the matrix \( A^4 B^3 \)**:
- Since the determinant of \( A^4 B^3 \) is zero, it implies that the matrix \( A^4 B^3 \) is a singular matrix.

### Final Answer:
The matrix \( A^4 B^3 \) is a singular matrix.