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Compute A^2 = A · A by direct multiplication.

The result of \( A^2 = A \cdot A \) is:

\[
A^2 = \begin{pmatrix} 
1 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 \\
0 & 0 & 1 & 0 \\
0 & 0 & 0 & 1 
\end{pmatrix}.
\]

This shows that the matrix \( A \) is idempotent, meaning that \( A^2 = A \).

Do you have any further questions or need more details?

### Related Questions:
1. What are the implications of a matrix being idempotent?
2. How would the matrix \( A^n \) behave for higher powers \( n \)?
3. What are some examples of other idempotent matrices in linear algebra?
4. How can we interpret the transpose operation geometrically?
5. Can we find the eigenvalues and eigenvectors of matrix \( A \)?

### Tip:
Idempotent matrices often have eigenvalues of either 0 or 1. This property can be useful in various applications such as projection matrices in linear algebra.

Explore how to compute A^2 = A · A for an idempotent matrix A. Learn about idempotent matrices and their properties. Suitable for advanced undergraduate students and those interested in matrix algebra.

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