https://file.aiquickdraw.com/mathbot/file/1730093272057fn0ma57e.jpg

To solve this problem, we need to find the eigenvalues of matrix \( A \), given that matrix \( P \) diagonalizes it. When a matrix \( A \) is diagonalized by a matrix \( P \), it means that:

\[
P^{-1} A P = D
\]

where \( D \) is a diagonal matrix with the eigenvalues of \( A \) on the diagonal.

### Given Data

Matrix \( A \) is:
\[
A = \begin{pmatrix} 15 - 6x & -4 + 2x \\ 42 - 21x & -11 + 7x \end{pmatrix}
\]

Matrix \( P \) is:
\[
P = \begin{pmatrix} -2 & 1 \\ -7 & 3 \end{pmatrix}
\]

### Steps to Solve

1. **Compute the Eigenvalues of \( A \):**
   Since \( P \) diagonalizes \( A \), we can compute \( D \) as \( D = P^{-1} A P \). The diagonal entries of \( D \) will be the eigenvalues of \( A \).

2. **Find \( P^{-1} \):**
   Calculate the inverse of \( P \), \( P^{-1} \), which is needed for finding \( D \).

3. **Multiply to Find \( D \):**
   Calculate \( D = P^{-1} A P \) and observe the resulting diagonal elements, which will be the eigenvalues of \( A \).

Let's proceed with these calculations.The eigenvalues of matrix \( A \) are:

\[
x + 1 \quad \text{and} \quad 3
\]

These can be written in any order as:

\[
x + 1; 3
\]

Would you like further details on this solution or have any additional questions?

### Related Questions
1. How can we verify if a matrix is diagonalizable?
2. What is the geometric interpretation of eigenvalues and eigenvectors?
3. How does changing \( x \) affect the eigenvalues of matrix \( A \)?
4. How are eigenvalues used in real-world applications like physics or engineering?
5. What would happen if \( P \) were not invertible?

### Tip
When diagonalizing a matrix, ensure that the matrix of eigenvectors \( P \) is invertible; otherwise, diagonalization is not possible.

Suppose that the matrix A = \( \begin{pmatrix} 15 - 6x & -4 + 2x \\ 42 - 21x & -11 + 7x \end{pmatrix} \) is diagonalised by P = \( \begin{pmatrix} -2 & 1 \\ -7 & 3 \end{pmatrix} \). Enter the eigenvalues of A in any order, separated by a semicolon.

Learn how to find the eigenvalues of a matrix A, which is diagonalized by matrix P. This problem covers key linear algebra concepts, including matrix diagonalization and eigenvalue computation. Suitable for introductory linear algebra students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation