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 $A$: 8 & 1 & -4 \\ 0 & 3 & 6 \\ 0 & 0 & 10 \end{bmatrix}$$
• Matrix $B$: 9 & 10 & 0 \\ 0 & 3 & 6 \\ 0 & 0 & -5 \end{bmatrix}$$

Matrix Products:

1. Product $AB$:

   • From the image, the result of $AB$ is: 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: 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.