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.