Matrix Transformation

To solve this problem, we need to determine the number of 3-tuples that satisfy the given equation. The equation involves determining the determinant of the adjugate matrix applied multiple times.

First, note the properties of the matrix:

• Matrix is: .
• Since are distinct natural numbers, is a square matrix.

To solve, we consider:

1. The determinant of , , is a polynomial in .
2. Using the properties of determinants and adjugate matrices:
   • for a matrix.
   • Repeating the adjugate operation won't change the degree of modulus transformation.
3. The given equation simplifies based on the properties of determinants and adjugates:
   • .
4. Therefore, the equation reduces to:
   • .
   • This implies , where .
5. Solve for : Since , .

This indicates .

Now, consider possible values for distinct tuples :

• Each permutation of three distinct numbers has a determinant satisfying the criterion.

Given no further constraints (e.g., values of ), we count permutations that maintain distinctness:

1. permutations for any distinct set of integers.

Finally, verify the solution range:

• The number of distinct 3-tuples is , which fits the range (42, 42) given.

Thus, the number of such 3-tuples is .

Step 1: Express in Trigonometric Form
We can rewrite as:

Here, .
Step 2: Powers of
For a rotation matrix , we know:

Calculate and :

Since and , we get:

Similarly:

From periodicity ( and ), we find:

Step 3: Verify the Relation
From the options:

Substitute and :

This is true.
Conclusion:

The correct answer is (C) : 4
Let A

⇒ p² +qr=1 (1) pq + qs = 0
⇒ q(p+s) = 0 (3)
⇒ s² + qr =1 (2) pr + rs = 0
⇒ r(p+s) = 0 (4)
From , eqn (1) - eqn (2)
p² = s² ⇒ p+s=0
Now 3a² + 4b²
= 3(p+s)² + 4(ps-qr)
= 3.0 + 4(-p²-qr)²
= 4(p² + qr )²
= 4

We are given the matrix , and we need to calculate the value of in the equation .
Step 1: Find the determinant of .
From the given matrix , we calculate the determinant : Step 2: Use the properties of the adjugate matrix.
We know the following properties of the adjugate matrix: Therefore, Step 3: Simplify the equation.
We can now solve the equation: Simplifying further: Final Answer: .

Given:


From this, we get the equation:


From the equation above, we derive:


Now solving for :


Therefore:


Finally, we calculate:


The final answer is 14.

Step 1: Understand the problem.
We are given the matrices , , and the expression . Additionally, we need to compute and find the value of , where is the result of the matrix multiplication.
Step 2: Simplify the expression for .
We start by calculating . First, we compute , the transpose of : Now, we compute : First, calculate : Next, multiply by : After performing the matrix multiplication, we find: Step 3: Compute .
Now, we compute . Since is the identity matrix, we have . Thus: We compute : Step 4: Compute .
The matrix , so we have , , , and . Now, calculate: Final Answer: 2005.