An `nxxn` matrix is formed using 0,1 and -1 as its elements. The number of such matrices which are skew-symmetric, is

Options

A. `(n(n+1))/2`
B. `(n-1)^2`
C. `2^((n(n-1))/(2))`
D. `3^((n(n-1))/(2))`

Explanation

To find the number of skew-symmetric matrices of order \( n \) formed using the elements \( 0, 1, \) and \( -1 \), we can follow these steps:

### Step 1: Understanding Skew-Symmetric Matrices
A matrix \( A \) is skew-symmetric if \( A^T = -A \). This implies that the diagonal elements of a skew-symmetric matrix must be zero, i.e., \( a_{ii} = 0 \) for all \( i \).

### Step 2: Counting the Off-Diagonal Elements
For an \( n \times n \) skew-symmetric matrix, the elements above the diagonal determine the elements below the diagonal. Specifically, if \( a_{ij} \) is an element above the diagonal, then \( a_{ji} = -a_{ij} \).

### Step 3: Identifying the Number of Off-Diagonal Elements
The total number of elements in an \( n \times n \) matrix is \( n^2 \). The diagonal elements contribute \( n \) elements (which are all 0), leaving us with \( n^2 - n \) off-diagonal elements. Since each off-diagonal pair \( (a_{ij}, a_{ji}) \) can be chosen independently, we only need to consider the elements above the diagonal.

The number of pairs of indices \( (i, j) \) where \( i < j \) is given by the combination \( \binom{n}{2} = \frac{n(n-1)}{2} \).

### Step 4: Choosing Values for Each Off-Diagonal Element
Each of these \( \frac{n(n-1)}{2} \) elements can take one of three values: \( 0, 1, \) or \( -1 \). Therefore, for each of these positions, we have 3 choices.

### Step 5: Total Number of Skew-Symmetric Matrices
The total number of skew-symmetric matrices can be calculated by raising the number of choices (3) to the power of the number of independent off-diagonal elements:

\[ \text{Total Matrices} = 3^{\frac{n(n-1)}{2}} \]

### Final Answer
Thus, the number of \( n \times n \) skew-symmetric matrices formed using the elements \( 0, 1, \) and \( -1 \) is:

\[ 3^{\frac{n(n-1)}{2}} \]

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