To prove that the product of upper triangular matrices is upper triangular, one effective method involves expressing the entries of the product matrix as dot products of rows from the first matrix and columns from the second matrix. For upper triangular matrix A, the elements satisfy the condition that a_{ij} = 0 when i > j. When multiplying two upper triangular matrices A and B to get C, it is necessary to demonstrate that c_{ij} = 0 for i > j. This approach simplifies the proof by focusing on the structure of the matrices rather than listing all entries. Ultimately, this method provides a clearer and more efficient way to establish the desired result.
