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Answer :
U1 + U2, U1U2, U^2, U1 + L1, and L1 + L2 are triangular matrices, while U1L1 may not be triangular. Among the given expressions, the following are triangular matrices: U1 + U2, U1U2, U^2, U1 + L1, and L1 + L2.
1. These matrices maintain the triangular structure because the addition or multiplication of two upper or lower triangular matrices preserves the triangular pattern. However, U1L1 is not necessarily triangular as the product of an upper triangular matrix and a lower triangular matrix can result in a matrix that is not triangular.
2. A triangular matrix is a special type of square matrix where all the elements either above or below the main diagonal are zero. In the given options, U1 + U2 and U1 + L1 are triangular matrices. When adding two upper triangular matrices, the resulting matrix will also be upper triangular as the sum of any two zero elements is still zero. Similarly, adding an upper and lower triangular matrix will result in an upper triangular matrix because the lower triangular part remains zero.
3. U1U2 represents the product of two upper triangular matrices, and the result is also an upper triangular matrix. Multiplication of two triangular matrices preserves the triangular pattern since the multiplication of zero with any element is zero, maintaining the upper triangular structure.
4. U^2 represents the square of an upper triangular matrix, and the resulting matrix is also upper triangular. Squaring an upper triangular matrix preserves the triangular pattern as each element in the original matrix is multiplied by itself.
5. On the other hand, U1L1 does not necessarily produce a triangular matrix. The product of an upper triangular matrix and a lower triangular matrix can result in a matrix where nonzero elements appear above the main diagonal, breaking the triangular pattern.
6. Lastly, L1 + L2 represents the addition of two lower triangular matrices, and the resulting matrix is also lower triangular. Adding two lower triangular matrices preserves the triangular structure, as the sum of any two zero elements is still zero.
7. In summary, U1 + U2, U1U2, U^2, U1 + L1, and L1 + L2 are triangular matrices, while U1L1 may not be triangular.
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