Question

An \( n \times n \) matrix is formed using 0, 1 and -1 as its elements. The number of such matrices which are skew symmetric is:

• A. \( \frac{n(n-1)}{2} \)
• B. \( (n-1)^2 \)
• C. \( 2^{n(n-1)/2} \)
• D. \( 3^{n(n-1)/2} \)

Hint:

For skew symmetric matrices, set diagonal elements to 0 and use the condition \( a_{ij} = -a_{ji} \). Count the independent off-diagonal elements (\( \frac{n(n-1)}{2} \)) and compute the number of choices for each.

Answer 1

The Correct Option is D

Step 1: Understand the definition of a skew symmetric matrix.
A matrix \( A \) is skew symmetric if \( A^T = -A \), where \( A^T \) is the transpose of \( A \). For an \( n \times n \) matrix, this implies \( a_{ij} = -a_{ji} \) for all \( i, j \), and the diagonal elements \( a_{ii} = 0 \) (since \( a_{ii} = -a_{ii} \) implies \( 2a_{ii} = 0 \)).
Step 2: Determine the number of independent elements.
In a skew symmetric matrix:
The diagonal elements \( a_{ii} \) (for \( i = 1 \) to \( n \)) must be 0.
For the off-diagonal elements \( a_{ij} \) (where \( i \neq j \)), \( a_{ij} = -a_{ji} \), so each pair \( (a_{ij}, a_{ji}) \) is determined by a single value \( a_{ij} \) (with \( a_{ji} = -a_{ij} \)).
The number of unique off-diagonal pairs \( (i, j) \) where \( i<j \) is the number of ways to choose 2 distinct indices from \( n \), which is:
\[ \binom{n}{2} = \frac{n(n-1)}{2}. \]
Step 3: Assign values using 0, 1, and -1.
Each independent off-diagonal element \( a_{ij} \) (for \( i<j \)) can be 0, 1, or -1, and the corresponding \( a_{ji} = -a_{ij} \):
If \( a_{ij} = 0 \), then \( a_{ji} = 0 \).
If \( a_{ij} = 1 \), then \( a_{ji} = -1 \).
If \( a_{ij} = -1 \), then \( a_{ji} = 1 \).
Thus, each pair \( (i, j) \) with \( i<j \) has 3 choices for \( a_{ij} \): 0, 1, or -1. The total number of independent pairs is \( \frac{n(n-1)}{2} \), and with 3 choices per pair, the total number of skew symmetric matrices is: \[ 3^{\frac{n(n-1)}{2}}. \]
Step 4: Verify the result with a small example.
For \( n = 1 \), the matrix is \( [a_{11}] \), and \( a_{11} = 0 \), so there is \( 3^{1(1-1)/2} = 3^0 = 1 \) matrix.
For \( n = 2 \), the matrix is: \[ \begin{bmatrix} 0 & a_{12}
-a_{12} & 0 \end{bmatrix}, \] where \( a_{12} \) can be 0, 1, or -1, giving \( 3^{2(2-1)/2} = 3^1 = 3 \) matrices, which matches.
Step 5: Select the correct answer.
The number of skew symmetric matrices is \( 3^{\frac{n(n-1)}{2}} \), which matches option (D).