Two eigenvalues of eigenvalues of A ^ 2 A = [[4, 6, 6], [1, 3, 2], [- 1, - 5, - 2]] equal and they are double of the third. Find the

Let's solve the problem step by step. We are asked to find the eigenvalues of the matrix \( A^2 \), where \( A \) is given as follows:

\[
A = \begin{bmatrix} 
4 & 6 & 6 \\ 
1 & 3 & 2 \\ 
-1 & -5 & -2 
\end{bmatrix}
\]

Additionally, we are told that two eigenvalues of \( A^2 \) are equal, and they are double the third eigenvalue.

### Steps:

1. **Find the matrix \( A^2 \):**

   To find \( A^2 \), we need to multiply matrix \( A \) by itself:
   \[
   A^2 = A \cdot A
   \]
   Let's compute \( A^2 \).

2. **Find the eigenvalues of \( A^2 \):**

   Once we have \( A^2 \), we will compute its eigenvalues. The condition given is that two eigenvalues are equal, and they are double the third eigenvalue. We'll use this information to find the eigenvalues.

Let me calculate these steps for you.The matrix \( A^2 \) is:

\[
A^2 = \begin{bmatrix} 
16 & 12 & 24 \\ 
5 & 5 & 8 \\ 
-7 & -11 & -12 
\end{bmatrix}
\]

Now, let's compute the eigenvalues of this matrix and apply the condition that two eigenvalues are equal, and they are double the third eigenvalue.It seems like I can’t do more advanced data analysis right now. Please try again later. However, you can follow the steps I mentioned to find the eigenvalues of the matrix \( A^2 \) using any math software or calculator. If you'd like, I can guide you through the manual steps to solve this!

Would you like me to explain how to calculate eigenvalues manually or assist with another query? 

Here are 5 related questions to further explore this topic:
1. How are eigenvalues of a matrix related to its determinant and trace?
2. What are some applications of eigenvalues and eigenvectors in physics?
3. How do you find the eigenvalues of a matrix without using a calculator?
4. What properties of eigenvalues remain unchanged when squaring a matrix?
5. How can you determine if a matrix is diagonalizable based on its eigenvalues?

**Tip**: When working with eigenvalues, remember that symmetric matrices always have real eigenvalues, which can simplify the process.

Two eigenvalues of eigenvalues of A^2 A = [[4, 6, 6], [1, 3, 2], [-1, -5, -2]] are equal and they are double of the third. Find the eigenvalues.

Solve for the eigenvalues of A^2 for a 3x3 matrix, where two eigenvalues are equal and double the third. The problem involves matrix multiplication, eigenvalue equations, and eigenvector theory. Suitable for university-level students studying linear algebra.

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