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The cube of the given matrix \( A \) is:

\[
A^3 = \begin{bmatrix}
-1 & 27 & -9 \\
18 & -28 & 0 \\
0 & 9 & -1
\end{bmatrix}
\]

Would you like a detailed explanation of how matrix exponentiation works or more details on this specific computation?

Here are five related questions to expand on this topic:
1. How would you compute higher powers of a matrix, like \( A^5 \)?
2. What is the significance of the determinant of a matrix raised to a power?
3. How do eigenvalues and eigenvectors behave under matrix exponentiation?
4. How can we use matrix powers in solving systems of linear equations?
5. What is the role of matrix powers in Markov chains?

**Tip:** Matrix exponentiation can be simplified using diagonalization if the matrix is diagonalizable.

Cube the given matrix: \[ \begin{bmatrix} 1 & 3 & -1 \\ 2 & -2 & 0 \\ 0 & 1 & 1 \end{bmatrix}^3 \]

Learn how to compute the cube of a 3x3 matrix \[ \begin{bmatrix} 1 & 3 & -1 \\ 2 & -2 & 0 \\ 0 & 1 & 1 \end{bmatrix} \]. This detailed solution walks through the matrix multiplication process and highlights key linear algebra concepts such as matrix powers and matrix multiplication properties.

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