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The image shows a matrix multiplication problem involving upper triangular matrices \( A \) and \( B \). The matrices are defined, and the task is to find their products \( A^2 \) and \( AB \), and determine whether these products are upper triangular.

### Definitions:
- **Upper triangular matrix**: A matrix is upper triangular if all the elements below the main diagonal are zero, i.e., \( a_{ij} = 0 \) for \( i > j \).

### Given Matrices:
- Matrix \( A \):
  \[
  A = \begin{bmatrix}
  8 & 1 & -4 \\
  0 & 3 & 6 \\
  0 & 0 & 10
  \end{bmatrix}
  \]
- Matrix \( B \):
  \[
  B = \begin{bmatrix}
  9 & 10 & 0 \\
  0 & 3 & 6 \\
  0 & 0 & -5
  \end{bmatrix}
  \]

### Matrix Products:
1. **Product \( AB \)**:
   - From the image, the result of \( AB \) is:
     \[
     AB = \begin{bmatrix}
     0 & 0 & 72 \\
     0 & 27 & 89 \\
     90 & 39 & -41
     \end{bmatrix}
     \]
   This matrix is **not upper triangular** because it has non-zero elements below the diagonal (specifically, 90 and 39).

2. **Product \( A^2 \)**:
   - From the image, the result of \( A^2 \) is:
     \[
     A^2 = \begin{bmatrix}
     64 & 0 & 11 \\
     0 & 9 & 0 \\
     100 & 78 & 0
     \end{bmatrix}
     \]
   This matrix is also **not upper triangular**, as the entries below the diagonal are not zero (e.g., 100 and 78).

### Conclusion:
Both products, \( AB \) and \( A^2 \), are **not upper triangular**. The calculations suggest that only \( A \) and \( B \) themselves are upper triangular, but their products do not maintain this property.

Would you like a detailed explanation of the multiplication process? Here are some related questions:

1. What are the conditions for the product of two upper triangular matrices to also be upper triangular?
2. Can a matrix product ever be upper triangular if one of the factors is not?
3. How does the structure of matrix \( A \) influence the result of \( A^2 \)?
4. What properties do upper triangular matrices maintain under addition and multiplication?
5. How are determinants and eigenvalues of triangular matrices calculated?

**Tip**: When multiplying matrices, check the structure of the product to ensure it maintains properties like being triangular or symmetric.

Find the products A^2 and AB. Are these products upper triangular?

Learn how to calculate the matrix products A^2 and AB for given upper triangular matrices A and B. Explore whether the resulting matrices are also upper triangular and understand the conditions under which the product of two upper triangular matrices remains upper triangular.

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