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 the nature of the square matrix \( A \) defined by the elements \( a_{ij} = (i - j)^3 \), we will analyze the properties of this matrix step by step.

### Step 1: Define the matrix elements
The elements of the matrix \( A \) are given by:
\[ a_{ij} = (i - j)^3 \]
This means that the element in the \( i \)-th row and \( j \)-th column is the cube of the difference between the row index \( i \) and the column index \( j \).

### Step 2: Check the property of skew-symmetry
A matrix \( A \) is said to be skew-symmetric if it satisfies the condition:
\[ A^T = -A \]
where \( A^T \) is the transpose of matrix \( A \). This means that for all \( i \) and \( j \):
\[ a_{ij} = -a_{ji} \]

### Step 3: Calculate \( a_{ji} \)
Using the definition of the matrix elements, we can find \( a_{ji} \):
\[ a_{ji} = (j - i)^3 \]
Now, we can express \( a_{ji} \) in terms of \( a_{ij} \):
\[ a_{ji} = (j - i)^3 = - (i - j)^3 = -a_{ij} \]
This shows that:
\[ a_{ij} = -a_{ji} \]

### Step 4: Conclusion about the matrix
Since we have established that \( a_{ij} = -a_{ji} \) for all \( i \) and \( j \), we can conclude that the matrix \( A \) is skew-symmetric.

### Final Answer
Thus, the square matrix \( A \) defined by \( a_{ij} = (i - j)^3 \) is a **skew-symmetric matrix**.

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