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The matrix \( A \) is given as:

\[
A = \begin{bmatrix} 4 & 0 & -1 \\ 0 & 1 & -4 \\ 0 & 0 & 0 \end{bmatrix}
\]

To find the eigenvalues, we solve the characteristic equation:

\[
\text{det}(A - \lambda I) = 0
\]

where \( I \) is the identity matrix and \( \lambda \) is the eigenvalue.

### Step 1: Set up the characteristic matrix
\[
A - \lambda I = \begin{bmatrix} 4 - \lambda & 0 & -1 \\ 0 & 1 - \lambda & -4 \\ 0 & 0 & -\lambda \end{bmatrix}
\]

### Step 2: Calculate the determinant

The determinant is:

\[
\text{det}(A - \lambda I) = (4 - \lambda)\begin{vmatrix} 1 - \lambda & -4 \\ 0 & -\lambda \end{vmatrix} = (4 - \lambda)((1 - \lambda)(-\lambda) - 0)
\]
\[
= (4 - \lambda)(-\lambda + \lambda^2)
= -\lambda(4 - \lambda)(\lambda - 1)
\]

### Step 3: Solve for eigenvalues

Set the determinant equal to zero:

\[
-\lambda(4 - \lambda)(\lambda - 1) = 0
\]

This gives us the solutions:

\[
\lambda = 0, \lambda = 4, \lambda = 1
\]

### Conclusion:

The eigenvalues are \( 0, 1, 4 \), corresponding to the first answer option: **0, 1, 4**.

Would you like further details or explanations?

Here are 5 related questions:
1. What is the geometric meaning of eigenvalues?
2. How do we find eigenvectors for a matrix once we have the eigenvalues?
3. What does the determinant of a matrix tell us in relation to its eigenvalues?
4. How do eigenvalues relate to the stability of a system in differential equations?
5. Can a matrix have complex eigenvalues? Under what conditions?

**Tip**: When calculating eigenvalues, always check if the matrix has any obvious properties (e.g., triangular form) to simplify the process.

The eigenvalues of the matrix A = [[4, 0, -1], [0, 1, -4], [0, 0, 0]] are asked in a multiple-choice format.

Explore how to find the eigenvalues of a 3x3 matrix A = [[4, 0, -1], [0, 1, -4], [0, 0, 0]]. This detailed explanation covers key concepts in linear algebra such as the characteristic equation and the calculation of determinants.

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