Final answer:
No, the sum of two lower triangular matrices cannot result in a non-lower triangular matrix because the elements above the diagonal in both matrices are zero, and the addition of zero with zero is zero, hence preserving the lower triangular structure.
Step-by-step explanation:
The question asks whether the sum of two lower triangular matrices can result in a non-lower triangular matrix. To answer this, let's recall what a lower triangular matrix is. A lower triangular matrix is one where all the entries above the main diagonal are zero. For example:
| a11 0 0 |
| a21 a22 0 |
| a31 a32 a33|
When you add two lower triangular matrices, each element in the resulting matrix will be the sum of the corresponding elements from the two original matrices. Since the elements above the diagonal in both matrices are zero, their sum would also be zero. Therefore, the sum will maintain the lower triangular form. Here's an example with two lower triangular matrices A and B:
Matrix A: Matrix B: Sum (A+B):
| a 0 0 | + | x 0 0 | = | a+x 0 0 |
| b c 0 | | y z 0 | | b+y c+z 0 |
| d e f | | u v w | | d+u e+v f+w |
As we can see, the (i, j) element where i < j (above the diagonal) in the sum matrix is 0+0=0, which means the sum matrix is still a lower triangular matrix. Hence, it is not possible for the sum of two lower triangular matrices to be a non-lower triangular matrix.