the square matrix `A=[a_(ij)]_mxxm` given by`a_(ij)=(i-j)^(n),` show that A is symmetic and skew-sysmmetic matrices according as n is even or odd, repectively.

Explanation

To show that the square matrix \( A = [a_{ij}]_{m \times m} \) defined by \( a_{ij} = (i - j)^n \) is symmetric when \( n \) is even and skew-symmetric when \( n \) is odd, we will follow these steps:

### Step 1: Define the conditions for symmetry and skew-symmetry
A matrix \( A \) is symmetric if \( A = A^T \) (i.e., \( a_{ij} = a_{ji} \) for all \( i, j \)). A matrix \( A \) is skew-symmetric if \( A = -A^T \) (i.e., \( a_{ij} = -a_{ji} \) for all \( i, j \)).

### Step 2: Analyze the elements of the matrix
Given \( a_{ij} = (i - j)^n \), we need to compare \( a_{ij} \) and \( a_{ji} \):
\[
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