The square matrix `A=[a_(ij)" given by "a_(ij)=(i-j)^3`, is a

Options

A. symmetric matrix
B. skew-symmetric matrix
C. diagonal matrix
D. hermitian matrix

Explanation

To determine whether the square matrix \( A = [a_{ij}] \) given by \( a_{ij} = (i - j)^3 \) is a skew-symmetric matrix, we will follow these steps:

### Step 1: Define the matrix elements
The elements of the matrix \( A \) are defined as:
\[ a_{ij} = (i - j)^3 \]

### Step 2: Find the transpose of the matrix
The transpose of matrix \( A \), denoted as \( A^T \), is defined as:
\[ a_{ji} = (j - i)^3 \]

### Step 3: Relate \( a_{ij} \) and \( a_{ji} \)
We can express \( a_{ji} \) in terms of \( a_{ij} \):
\[ a_{ji} = (j - i)^3 = -(i - j)^3 = -a_{ij} \]
This shows that:
\[ A^T = -A \]

### Step 4: Conclusion about the matrix
Since \( A^T = -A \), we conclude that the matrix \( A \) is skew-symmetric.

### Final Answer
The square matrix \( A \) is a **skew-symmetric matrix**.

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