Matrices

Given matrices:

, ,

Since is an orthogonal matrix, we must have , where is the identity matrix. Therefore:

This implies that and , so and .

We are given that . Since is orthogonal, . Thus, .

Comparing the matrices, we have:

From this, we can see that . Since can be either 1 or -1 but it does not affect the final multiplication, we can consider two possible values: If and , then . If and , then . Since x must be 1 and y does not appear in the final matrix equation, we have B=PAP^{-1}.

Thus, , and can be either or . However, from the given options, the most suitable answer is and . This is because only x affects our result and y is arbitrary due to the zero rows and columns.

Given: A matrix

Step 1: Compute the determinant of

Step 2: Sum of determinants from to

This is an arithmetic series with first term and last term .

Number of terms = 2008

Now, note that:

Final Answer: Option (C):