Matrices

To solve the problem, we start by analyzing the given conditions about the square matrix .

1. The matrix satisfies the condition . This means is an idempotent matrix. Idempotent matrices have the property that their eigenvalues can only be 0 or 1.
2. We are also given that . This equation holds for some scalar .
3. To find , let's expand using the binomial theorem: Given the property of the identity matrix and the idempotent property for , the expression simplifies to: but more generally considering ,
4. Recognizing that: (because and ), we equate this to the form in the problem : Therefore, .

Thus, the value of is .

Step 1: Understanding the Concept:
Substitute to simplify the entries, then check for a skew-symmetric structure.
Step 2: Detailed Explanation:
and .
The matrix becomes , which is skew-symmetric of odd order (3).
The determinant of any skew-symmetric matrix of odd order is .
Step 3: Final Answer:
The determinant equals .