Matrix

Let's solve the given problem step-by-step.

Given that , matrix is orthogonal.

We need to find the value of:

First, we expand the expression :

Since and , we have:

• , thus

Similarly, expand the expression :

Adding both expansions, we get:

The terms and cancel each other, so:

Substituting back into the original expression:

Now simplify:

Since for orthogonal matrices:

The given problem aims to simplify to reveal a simpler form, so we double-check another route:

(since ) aligns to the option

Thus, the correct answer is:

A³+A^T

We are given the matrix . We need to determine the value of when .

Firstly, let's understand the relationship between a matrix , its adjugate (denoted ), and the determinant:

• The determinant of the adjugate of is given by where is the order of the matrix .
• If is a matrix, then .
• For any scalar , .

Now, let's solve the problem step-by-step:

1. Determine the determinant of matrix . The determinant of a scalar multiplied by a matrix is given by: .
2. Since for matrix :
3. Next, compute using :
4. To find , we use:
5. Solving , we have:
6. Setting , implies and . Plugging this into the above expression:
7. This solution yields , . Summing up, we find:

Thus, the correct option is 24.

Step 1: Understand the condition . If for all nonzero vectors , then must be a skew-symmetric matrix. Therefore, .

Step 2: Use the given information to find . Let and . We are given:

Also, let be an arbitrary vector. Since is skew-symmetric, the diagonal entries must be zero.

Let us compute :

Since we do not know enough about , let us proceed. The relation for all implies is skew-symmetric. Let , then: From , we know for all . Let . Then: and So we have the following system of equations: From and , we get and . From , we get . Therefore, the matrix is:

Step 3: Calculate

We first compute , where is the identity matrix: Then: The determinant of is calculated as:

Step 4: Calculate

We know that: where is the size of the matrix. In our case, , so:

Step 5: Find and

We have: so , , and . Thus:

Final Answer:

The value of is .