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) is it possible that ""the sum of two lower triangular matrices be non-lower triangular matrix"" ? explain.

User Joelvh
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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.

User Guillaume Boudreau
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4 votes

Final answer:

The sum of two lower triangular matrices is always a lower triangular matrix because when adding the corresponding entries, the upper triangular part of the sum will remain zero.

Step-by-step explanation:

The question whether "the sum of two lower triangular matrices can be a non-lower triangular matrix" refers to a concept in linear algebra. In the context of matrices, a lower triangular matrix is one where all the elements above the diagonal are zero. When two lower triangular matrices are added, each corresponding entry in the sum is the addition of the two entries from the original matrices. Therefore, the upper triangular part of the sum will remain zero since adding zeros together still results in zero. Consequently, the sum of two lower triangular matrices is always a lower triangular matrix.

User Karim Tarek
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