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!

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.