https://file.aiquickdraw.com/mathbot/file/1726911164375l3mvhpo7.jpg

The matrix given in the image is:

\[
\begin{pmatrix}
3 & 10 & 5 \\
-2 & -3 & -4 \\
3 & 5 & 7
\end{pmatrix}
\]

It looks like the instruction is to find the characteristic polynomial of the matrix. The characteristic polynomial of a matrix \( A \) is given by:

\[
\text{det}(A - \lambda I)
\]

Where:
- \( A \) is the matrix,
- \( I \) is the identity matrix of the same dimension,
- \( \lambda \) is a scalar, and
- \( \text{det} \) represents the determinant.

Let's go step by step:
1. Compute the matrix \( A - \lambda I \).
2. Find the determinant of \( A - \lambda I \) to get the characteristic polynomial.

I will compute this for you.The characteristic polynomial of the given matrix is:

\[
\lambda^3 - 7\lambda^2 + 16\lambda - 12
\]

This polynomial is essential for finding eigenvalues and further analysis of the matrix.

Would you like to explore any details or have further questions? Here are some related queries:

1. How do you find the eigenvalues from the characteristic polynomial?
2. What does the characteristic polynomial tell us about the matrix?
3. Can you show how to compute the determinant of a matrix?
4. What are the properties of a matrix based on its eigenvalues?
5. How does the characteristic polynomial relate to diagonalization?

**Tip:** Eigenvalues are the roots of the characteristic polynomial, and they provide crucial information about the behavior of linear transformations defined by the matrix.

Find the characteristic polynomial of the matrix given below:

[
(3, 10, 5),
(-2, -3, -4),
(3, 5, 7)
]

Learn how to find the characteristic polynomial for a 3x3 matrix, covering key concepts like determinants, eigenvalues, and linear algebra. This problem demonstrates how to use the formula det(A - λI) to solve the matrix step by step.

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