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.