To verify the Cayley-Hamilton theorem for a given matrix $A$, we'll follow these steps:

1. Calculate the characteristic polynomial of $A$.
2. Substitute $A$ into its own characteristic polynomial.
3. Show that the matrix expression equals the zero matrix, confirming the theorem.

From your input, it seems the matrix $A$ is partially given as follows: $A = \begin{pmatrix} -1 & 2 \\ 1 & 0 & 2 \end{pmatrix}$ However, this matrix is incomplete. Could you clarify the matrix dimensions and all its entries so I can proceed with the verification?

Once clarified, I'll calculate the characteristic polynomial, substitute $A$ into it, and check if it equals the zero matrix.

Would you like more details on any of these steps?

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Here are some related questions to consider:

1. What are the eigenvalues of matrix $A$ and how do they relate to the Cayley-Hamilton theorem?
2. Can the theorem be verified for any size of square matrix, or are there specific limitations?
3. How does the Cayley-Hamilton theorem help in computing matrix functions, like $A^{-1}$?
4. What role does the characteristic polynomial play in determining the properties of a matrix?
5. How can we apply the Cayley-Hamilton theorem to simplify powers of matrices?

Tip: The Cayley-Hamilton theorem is particularly useful for expressing matrix powers in terms of lower powers by leveraging the characteristic polynomial.