https://file.aiquickdraw.com/mathbot/file/1725537311460rmefvz83.jpg

The image shows a matrix \( A \) and a polynomial equation related to it. The matrix \( A \) is given as:

\[
A = \begin{pmatrix}
2 & -1 & 1 \\
-1 & 2 & 1 \\
1 & -1 & 2
\end{pmatrix}
\]

And the equation to prove is:

\[
A^3 - 6A^2 + 9A - 4I = 0
\]

Where \( I \) is the identity matrix of the same order as \( A \). This looks like an exercise in verifying the Cayley-Hamilton theorem or working with characteristic polynomials.

### Steps to Verify the Equation:

1. **Find \( A^2 \)**: First, calculate \( A^2 \) by multiplying \( A \) by itself.
   
2. **Find \( A^3 \)**: Then, multiply \( A^2 \) by \( A \) to get \( A^3 \).
   
3. **Substitute into the Equation**: Use the values of \( A^2 \) and \( A^3 \), and compute \( -6A^2 + 9A - 4I \).

4. **Check the Result**: The goal is to confirm that the left-hand side equals the zero matrix, verifying the equation.

Let me calculate each of these steps for you!The result of the expression \( A^3 - 6A^2 + 9A - 4I \) is:

\[
\begin{pmatrix}
-2 & 2 & -2 \\
2 & -2 & -2 \\
-2 & 2 & -2
\end{pmatrix}
\]

This is not the zero matrix, meaning the given equation \( A^3 - 6A^2 + 9A - 4I = 0 \) is not valid for the provided matrix \( A \). There might be a mistake in the problem, or a different characteristic polynomial could be relevant.

Would you like to explore the characteristic polynomial for this matrix to better understand its properties?

Here are 5 related questions:
1. What is the characteristic polynomial of matrix \( A \)?
2. How do you find eigenvalues and eigenvectors for matrix \( A \)?
3. Could a different matrix satisfy the equation \( A^3 - 6A^2 + 9A - 4I = 0 \)?
4. What is the Cayley-Hamilton theorem, and how does it apply to matrices?
5. Can we generalize this type of matrix equation for higher powers of \( A \)?

**Tip**: When verifying matrix equations, it's crucial to carefully compute each step and check if the given equation aligns with the matrix properties (like eigenvalues).

Prove that A^3 - 6A^2 + 9A - 4I = 0 for the matrix A = [[2, -1, 1], [-1, 2, 1], [1, -1, 2]].

Explore how to prove the matrix equation A^3 - 6A^2 + 9A - 4I = 0 for the given matrix A = [[2, -1, 1], [-1, 2, 1], [1, -1, 2]]. Learn about key matrix algebra concepts such as the Cayley-Hamilton theorem and matrix multiplication.

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