To solve this problem, let's analyze the requirements step-by-step.

Problem Overview

We need to find an invertible matrix $P$ and a matrix $C$ of the form $C = \begin{bmatrix} a & -b \\ b & a \end{bmatrix}$ such that for a given matrix $A = \begin{bmatrix} 1 & -1.7 \\ 0.4 & 0.6 \end{bmatrix}$, it has the form $A = PCP^{-1}$.

This is asking if we can find a similarity transformation that will make $A$ similar to a matrix $C$ with that particular form.

Solution Steps

1. Check for Matrix Similarity: For $A$ and $C$ to be similar, they must have the same eigenvalues. So, let's start by finding the eigenvalues of $A$.

2. Calculate Eigenvalues of $A$: The eigenvalues of $A$ can be found by solving the characteristic polynomial $\det(A - \lambda I) = 0$.

3. Characteristic Polynomial of $A$: $\det\left(\begin{bmatrix} 1 - \lambda & -1.7 \\ 0.4 & 0.6 - \lambda \end{bmatrix}\right) = 0$ Expanding this determinant gives us: $(1 - \lambda)(0.6 - \lambda) - (-1.7)(0.4) = 0$ Simplifying, we get: $\lambda^2 - 1.6\lambda + 1 = 0$ Solving this quadratic equation will yield the eigenvalues of $A$.

4. Form of Matrix $C$: The form of $C$ indicates that it should be a matrix with complex eigenvalues (since it represents a rotation matrix). The eigenvalues of a matrix of the form $\begin{bmatrix} a & -b \\ b & a \end{bmatrix}$ are $a \pm bi$, which are complex.

5. Conclusion: If the eigenvalues of $A$ are real, then it’s impossible to find a matrix $C$ of the specified form that $A$ would be similar to. Conversely, if the eigenvalues are complex, we might be able to find such matrices $P$ and $C$.

Solve for Eigenvalues

Let's go ahead and solve for the eigenvalues of $A$.

Eigenvalue Solution

The discriminant of the characteristic polynomial $\lambda^2 - 1.6\lambda + 1 = 0$ is: $1.6^2 - 4 \cdot 1 \cdot 1 = 2.56 - 4 = -1.44$ Since the discriminant is negative, the eigenvalues are complex.

Thus, Option A is feasible, and we can find matrices $P$ and $C$ with the specified form.

Answer

Therefore, the correct choice is: A. The matrices $P$ and $C$ exist.

Would you like me to proceed with finding the exact forms of $P$ and $C$, or do you have other questions?

Here are five related questions for further exploration:

1. How do we construct the matrix $P$ given the eigenvalues and eigenvectors?
2. What is the physical significance of the matrix form $\begin{bmatrix} a & -b \\ b & a \end{bmatrix}$?
3. Can we determine the eigenvectors of $A$ and use them to form $P$?
4. How would we proceed if $A$ had real eigenvalues instead?
5. How do complex eigenvalues affect the stability of a dynamical system?

Tip: When working with matrix similarity transformations, always verify eigenvalues first to ensure compatibility with the desired form.