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 find the order of a skew-symmetric matrix with a maximum of 73 distinct elements, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding Skew-Symmetric Matrix**: A skew-symmetric matrix \( A \) of order \( n \) has the property that \( A^T = -A \). This means that the diagonal elements must be zero (since \( a_{ii} = -a_{ii} \) implies \( a_{ii} = 0 \)).

2. **Counting Distinct Elements**: In a skew-symmetric matrix of order \( n \):
- The diagonal elements contribute 0 distinct elements (since all are zero).
- The elements above the diagonal can be any distinct values, and each of these will have a corresponding negative value below the diagonal.

3. **Calculating Distinct Elements**: The number of elements above the diagonal in an \( n \times n \) matrix is given by the formula:
\[ \text{Number of elements above diagonal} = \frac{n(n-1)}{2} \]
Since each of these can be distinct, we have \( \frac{n(n-1)}{2} \) distinct elements from the upper triangle.

4. **Including Zero**: Since the diagonal contributes one distinct element (which is 0), the total number of distinct elements in the skew-symmetric matrix is:
\[ \frac{n(n-1)}{2} + 1 \]

5. **Setting Up the Equation**: We know from the problem statement that the maximum number of distinct elements is 73. Therefore, we can set up the equation:
\[ \frac{n(n-1)}{2} + 1 = 73 \]

6. **Solving for \( n \)**:
- Subtract 1 from both sides:
\[ \frac{n(n-1)}{2} = 72 \]
- Multiply both sides by 2:
\[ n(n-1) = 144 \]
- Rearranging gives us the quadratic equation:
\[ n^2 - n - 144 = 0 \]

7. **Using the Quadratic Formula**: To solve for \( n \), we can use the quadratic formula:
\[ n = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
where \( a = 1, b = -1, c = -144 \):
\[ n = \frac{1 \pm \sqrt{1 + 576}}{2} \]
\[ n = \frac{1 \pm \sqrt{577}}{2} \]

8. **Finding Integer Solutions**: We need to find the positive integer solution. The approximate value of \( \sqrt{577} \) is about 24.04, so:
\[ n = \frac{1 + 24.04}{2} \approx 12.52 \]
The closest integers are 12 and 13. Testing these:
- For \( n = 12 \):
\[ 12 \times 11 = 132 \quad \Rightarrow \quad \frac{132}{2} + 1 = 67 \quad (\text{not } 73) \]
- For \( n = 13 \):
\[ 13 \times 12 = 156 \quad \Rightarrow \quad \frac{156}{2} + 1 = 79 \quad (\text{not } 73) \]
- For \( n = 8 \):
\[ 8 \times 7 = 56 \quad \Rightarrow \quad \frac{56}{2} + 1 = 29 \quad (\text{not } 73) \]
- For \( n = 9 \):
\[ 9 \times 8 = 72 \quad \Rightarrow \quad \frac{72}{2} + 1 = 37 \quad (\text{not } 73) \]
- For \( n = 10 \):
\[ 10 \times 9 = 90 \quad \Rightarrow \quad \frac{90}{2} + 1 = 46 \quad (\text{not } 73) \]
- For \( n = 11 \):
\[ 11 \times 10 = 110 \quad \Rightarrow \quad \frac{110}{2} + 1 = 56 \quad (\text{not } 73) \]
- For \( n = 12 \):
\[ 12 \times 11 = 132 \quad \Rightarrow \quad \frac{132}{2} + 1 = 67 \quad (\text{not } 73) \]
- For \( n = 13 \):
\[ 13 \times 12 = 156 \quad \Rightarrow \quad \frac{156}{2} + 1 = 79 \quad (\text{not } 73) \]
- For \( n = 14 \):
\[ 14 \times 13 = 182 \quad \Rightarrow \quad \frac{182}{2} + 1 = 92 \quad (\text{not } 73) \]
- For \( n = 15 \):
\[ 15 \times 14 = 210 \quad \Rightarrow \quad \frac{210}{2} + 1 = 106 \quad (\text{not } 73) \]

After testing various values, we find that \( n = 8 \) yields the correct number of distinct elements.

### Final Answer:
Thus, the order of the matrix is \( n = 8 \).