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Given the matrix \( A = \begin{pmatrix} 2 & 1 \\ -4 & -2 \end{pmatrix} \), we need to find the value of \( I - A + A^2 - A^4 \), where \( I \) is the identity matrix.

First, we calculate \( A^2 \):
\[ A^2 = A \cdot A = \begin{pmatrix} 2 & 1 \\ -4 & -2 \end{pmatrix} \cdot \begin{pmatrix} 2 & 1 \\ -4 & -2 \end{pmatrix} = \begin{pmatrix} 4 - 4 & 2 - 2 \\ -8 + 8 & -4 + 4 \end{pmatrix} = \begin{pmatrix} 0 & 0 \\ 0 & 0 \end{pmatrix} \]

Since \( A^2 = 0 \), \( A^4 \) will also be zero because multiplying the zero matrix by any matrix still results in the zero matrix:
\[ A^4 = (A^2)^2 = 0 \]

Now we substitute these into the expression:
\[ I - A + A^2 - A^4 = I - A + 0 - 0 = I - A \]

The identity matrix \( I \) for a 2x2 matrix is:
\[ I = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} \]

Thus, we have:
\[ I - A = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} - \begin{pmatrix} 2 & 1 \\ -4 & -2 \end{pmatrix} = \begin{pmatrix} 1 - 2 & 0 - 1 \\ 0 + 4 & 1 + 2 \end{pmatrix} = \begin{pmatrix} -1 & -1 \\ 4 & 3 \end{pmatrix} \]

Therefore, the value of \( I - A + A^2 - A^4 \) is:
\[ \begin{pmatrix} -1 & -1 \\ 4 & 3 \end{pmatrix} \]

Do you have any questions or would you like more details on any part of this problem?

Here are some related questions to consider:
1. What is the determinant of matrix \( A \)?
2. What is the inverse of matrix \( A \), if it exists?
3. How do you find the eigenvalues and eigenvectors of matrix \( A \)?
4. What does it mean for a matrix to be nilpotent?
5. How would you verify the result of \( I - A + A^2 - A^4 \) using a different method?
6. What is the trace of matrix \( A \)?
7. How can you use the characteristic polynomial to find powers of matrices?
8. How does the Cayley-Hamilton theorem apply to this problem?

**Tip:** When dealing with matrix exponentiation, verifying intermediate results can help ensure accuracy in complex calculations.

Learn how to calculate the expression I - A + A^2 - A^4 for the matrix A = [2 1; -4 -2]. This problem explores matrix algebra concepts, including matrix multiplication, subtraction, and the Cayley-Hamilton theorem. Suitable for advanced high school students and beyond.

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