https://file.aiquickdraw.com/mathbot/file/17263890778978bh2lem2.jpg

The equation shown in the image is:

\[
\det(A + xI) = 0
\]

This represents a **characteristic equation**, where \( A \) is a square matrix, \( I \) is the identity matrix of the same size as \( A \), and \( x \) is a scalar. The determinant of the matrix \( (A + xI) \) is equal to zero.

### Explanation:

1. The term \( A + xI \) means we are adding the scalar \( x \) to every diagonal element of the matrix \( A \). This is essentially shifting the eigenvalues of \( A \) by \( -x \).
   
2. The determinant being zero implies that \( A + xI \) is a singular matrix, meaning it has at least one zero eigenvalue.

3. Solving \( \det(A + xI) = 0 \) typically involves finding the values of \( x \) that make the matrix \( A + xI \) singular. These values of \( x \) are related to the **eigenvalues** of the matrix \( A \), where \( -x \) corresponds to an eigenvalue of \( A \).

Would you like to go deeper into any specific part of this equation or solve an example? Let me know!

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Here are some follow-up questions you might find helpful:

1. What is an eigenvalue, and how is it related to this determinant equation?
2. How do we find the eigenvalues of a matrix \( A \)?
3. Can you explain the properties of the identity matrix \( I \)?
4. What does it mean for a matrix to be singular?
5. How do we compute the determinant of a matrix in practice?

**Tip:** The determinant of a matrix gives insight into whether the matrix is invertible. A determinant of zero implies the matrix is singular, i.e., it does not have an inverse.

Explore the characteristic equation det(A + xI) = 0, which involves finding eigenvalues of a matrix A. Learn how to solve this equation by analyzing the determinant and how it relates to eigenvalues. Suitable for undergraduate-level students.

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