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The image presents a mathematical problem where a matrix \( A^3 \) is calculated as \( A^2 \cdot A \), with the matrix \( A \) provided.

### Given:
Matrix \( A \):
\[
A = 
\begin{bmatrix}
1 & 2 & -1 \\
\frac{7}{2} & -\frac{3}{2} & 1 \\
2 & 1 & -2
\end{bmatrix}
\]

The intermediate result \( A^2 \) is also shown in the image:
\[
A^2 = 
\begin{bmatrix}
5 & -5 & 3 \\
-4 & 16 & -8 \\
1 & -1 & 3
\end{bmatrix}
\]

To compute \( A^3 \), we multiply \( A^2 \) by \( A \). Let me calculate this.import numpy as np

# Define matrices A and A^2
A = np.array([
    [1, 2, -1],
    [7/2, -3/2, 1],
    [2, 1, -2]
])

A2 = np.array([
    [5, -5, 3],
    [-4, 16, -8],
    [1, -1, 3]
])

# Compute A^3 = A^2 * A
A3 = np.dot(A2, A)
A3array([[ -6.5,  20.5, -16. ],
       [ 36. , -40. ,  36. ],
       [  3.5,   6.5,  -8. ]])array([[ -6.5,  20.5, -16. ],
       [ 36. , -40. ,  36. ],
       [  3.5,   6.5,  -8. ]])successThe result of \( A^3 = A^2 \cdot A \) is:

\[
A^3 = 
\begin{bmatrix}
-6.5 & 20.5 & -16 \\
36 & -40 & 36 \\
3.5 & 6.5 & -8
\end{bmatrix}
\]

Would you like a detailed explanation of the matrix multiplication process? 

Here are some related questions to explore:
1. How is matrix multiplication defined in general?
2. What are the applications of matrix powers in real-world scenarios?
3. How can we compute the eigenvalues and eigenvectors of this matrix \( A \)?
4. Can we find the determinant and inverse of \( A \)?
5. What are the implications if matrix \( A \) is singular?

**Tip:** Always check if matrices are compatible for multiplication (i.e., the number of columns in the first matrix equals the number of rows in the second).

Calculate A^3 given A^3 = A^2 · A, where matrix A is provided.

Explore how to compute the cube of a matrix (A^3) using matrix multiplication. This detailed example demonstrates A^3 = A^2 · A, using a provided matrix A and intermediate calculations of A^2. Ideal for students learning Linear Algebra.

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