If matrix `A=[a_(ij)]_(3xx)`, 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 given conditions, and then find the value of \( A^4 B^3 \).

### 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 \( A^T = -A \).

**Hint**: Remember that a skew-symmetric matrix has the property that its diagonal elements are zero.

2. **Determine the determinant of matrix \( A \)**:
- It is known that the determinant of any skew-symmetric matrix of odd order (like our \( 3 \times 3 \) matrix \( A \)) is zero.
- Therefore, \( \text{det}(A) = 0 \).

**Hint**: For skew-symmetric matrices, if the order is odd, the determinant is always zero.

3. **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 \( B^T = B \).

**Hint**: A symmetric matrix has equal elements across the diagonal, i.e., \( b_{ij} = b_{ji} \).

4. **Calculate \( A^4 B^3 \)**:
- We want to find \( A^4 B^3 \).
- We can use the property of determinants: \( \text{det}(A^4 B^3) = \text{det}(A^4) \cdot \text{det}(B^3) \).
- Since \( \text{det}(A) = 0 \), we have \( \text{det}(A^4) = (\text{det}(A))^4 = 0^4 = 0 \).

**Hint**: The determinant of a product of matrices is the product of their determinants.

5. **Conclusion about \( A^4 B^3 \)**:
- Since \( \text{det}(A^4) = 0 \), it follows that \( \text{det}(A^4 B^3) = 0 \cdot \text{det}(B^3) = 0 \).
- A matrix with a determinant of zero is a singular matrix.

**Hint**: A singular matrix is one that does not have an inverse, which is indicated by a determinant of zero.

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