Question 9

Form the matrix

C whose left column is the vector

v1

and whose right column is

v2

from immediately above.

By calculating

D=C^-1 AC or by using another method, find the diagonal matrix D.

To solve this problem, let's break it down into steps:

1. **Form the Matrix \( C \):**
   - Given that matrix \( C \) is composed of the vectors \( v_1 \) and \( v_2 \), the left column of \( C \) will be \( v_1 \) and the right column will be \( v_2 \).
   - Suppose \( v_1 = \begin{bmatrix} a_1 \\ b_1 \end{bmatrix} \) and \( v_2 = \begin{bmatrix} a_2 \\ b_2 \end{bmatrix} \).
   - Then, the matrix \( C \) will be:
     \[
     C = \begin{bmatrix}
     a_1 & a_2 \\
     b_1 & b_2
     \end{bmatrix}
     \]

2. **Find the Inverse of \( C \) (i.e., \( C^{-1} \)):**
   - The inverse of a 2x2 matrix \( C = \begin{bmatrix} p & q \\ r & s \end{bmatrix} \) is given by:
     \[
     C^{-1} = \frac{1}{\text{det}(C)} \begin{bmatrix} s & -q \\ -r & p \end{bmatrix}
     \]
   - Where the determinant \( \text{det}(C) \) is \( ps - qr \).

3. **Calculate the Matrix \( D = C^{-1}AC \):**
   - To diagonalize matrix \( A \), we multiply it by \( C^{-1} \) on the left and by \( C \) on the right.
   - The result \( D \) will be the diagonal matrix where the diagonal elements are the eigenvalues of \( A \).

4. **Alternative Method:**
   - If we have the eigenvalues \( \lambda_1 \) and \( \lambda_2 \) of \( A \), the diagonal matrix \( D \) can also be directly written as:
     \[
     D = \begin{bmatrix}
     \lambda_1 & 0 \\
     0 & \lambda_2
     \end{bmatrix}
     \]

### Example Calculation (Hypothetical Values)
Let's assume specific values for \( v_1 \), \( v_2 \), and \( A \) for demonstration.

Suppose:
\[
v_1 = \begin{bmatrix} 1 \\ 2 \end{bmatrix}, \quad v_2 = \begin{bmatrix} 3 \\ 4 \end{bmatrix}, \quad A = \begin{bmatrix} 5 & 6 \\ 7 & 8 \end{bmatrix}
\]
Then:
\[
C = \begin{bmatrix} 1 & 3 \\ 2 & 4 \end{bmatrix}
\]
The determinant of \( C \):
\[
\text{det}(C) = (1)(4) - (3)(2) = 4 - 6 = -2
\]
The inverse \( C^{-1} \) is:
\[
C^{-1} = \frac{1}{-2} \begin{bmatrix} 4 & -3 \\ -2 & 1 \end{bmatrix} = \begin{bmatrix} -2 & 1.5 \\ 1 & -0.5 \end{bmatrix}
\]
Now, calculate \( D = C^{-1}AC \).

If you have specific vectors \( v_1 \) and \( v_2 \), and a matrix \( A \), I can calculate the exact matrix \( D \) for you.

Would you like to provide those values or do you have any questions? 

---

### Relative Questions:
1. How is the determinant of a matrix used in finding its inverse?
2. Can every matrix be diagonalized? Why or why not?
3. What are eigenvalues and eigenvectors, and how are they related to matrix diagonalization?
4. Why is the diagonalization of a matrix useful in mathematical analysis?
5. How do the values of \( v_1 \) and \( v_2 \) affect the matrix \( C \) and the resulting diagonal matrix \( D \)?

**Tip:** Diagonalizing a matrix is a powerful tool in simplifying matrix operations, particularly for finding powers of matrices or solving differential equations.

Learn how to calculate the diagonal matrix D from matrix C inversion. Step-by-step solution involving linear algebra concepts such as matrix inversion and diagonalization. Suitable for advanced mathematics students and professionals.

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