A skew - symmetric matrix of order n has the maximum number of distinct elements equal to 73, then the order of the matrix is

Options

A. 7
B. 8
C. 9
D. 10

Answer

To solve the problem, let's break it down step by step.

### Step 1: Understanding Skew-Symmetric Matrices
A skew-symmetric matrix \( A \) of order \( n \) has the property that \( A^T = -A \). This means that the diagonal elements of the matrix must be zero, and the elements above the diagonal are the negatives of the elements below the diagonal.

### Step 2: Counting Elements
For a skew-symmetric matrix of order \( n \):
- The diagonal elements contribute **1 distinct element** (which is 0).
- The elements above the diagonal can be filled with distinct elements, and each of these will have a corresponding negative element below the diagonal.

### Step 3: Total Number of Distinct Elements
The number of distinct elements in a skew-symmetric matrix can be calculated as follows:
- The number of elements above the diagonal in an \( n \times n \) matrix is given by \( \frac{n(n-1)}{2} \) (this is the combination of \( n \) elements taken 2 at a time).
- Including the diagonal element (which is 0), the total number of distinct elements in the matrix is:
\[ \text{Total distinct elements} = 1 + \frac{n(n-1)}{2} \]

### Step 4: Setting Up the Equation
According to the problem, the maximum number of distinct elements is given as 73. Therefore, we can set up the equation:
\[ 1 + \frac{n(n-1)}{2} = 73 \]

### Step 5: Simplifying the Equation
Subtract 1 from both sides:
\[ \frac{n(n-1)}{2} = 72 \]
Multiply both sides by 2:
\[ n(n-1) = 144 \]

### Step 6: Solving the Quadratic Equation
Now we need to solve the quadratic equation:
\[ n^2 - n - 144 = 0 \]
Using the quadratic formula \( n = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \):
Here, \( a = 1, b = -1, c = -144 \):
\[ n = \frac{-(-1) \pm \sqrt{(-1)^2 - 4 \cdot 1 \cdot (-144)}}{2 \cdot 1} \]
\[ n = \frac{1 \pm \sqrt{1 + 576}}{2} \]
\[ n = \frac{1 \pm \sqrt{577}}{2} \]

### Step 7: Finding Integer Solutions
Calculating \( \sqrt{577} \) gives approximately 24.08. Thus:
\[ n = \frac{1 + 24.08}{2} \approx 12.54 \quad \text{(not an integer)} \]
\[ n = \frac{1 - 24.08}{2} \quad \text{(not valid as it gives a negative)} \]

Instead, let's factor \( n(n-1) = 144 \):
The pairs that multiply to 144 are:
- \( (12, 12) \)
- \( (9, 16) \)
- \( (8, 18) \)
- \( (6, 24) \)
- \( (4, 36) \)

Among these, \( n = 12 \) works since \( 12 \times 11 = 132 \) and \( 12 \) is the only integer solution.

### Conclusion
Thus, the order of the skew-symmetric matrix is:
\[ \boxed{12} \]