**Properties of Triangular Matrices**1. **Lower Triangular Matrices Multiplication:** - **Statement:** Let $ L_1 $ and $ L_2 $ be two lower triangular $ n \times n $ matrices. Show that $ L_1L_2 $ is also lower triangular. - **Hint:** The $(i, j)$-th entry of $ L_1L_2 $ is the inner product of the $ i $-th row of $ L_1 $ with the $ j $-th column of $ L_2 $. If $ i < j $, that is, if the entry we compute is above the diagonal, show that the inner product in question is 0.2. **Upper Triangular Matrices Multiplication:** - **Conclusion:** The product of two upper triangular $ n \times n $ matrices is also upper triangular (e.g., by using transposition, and part (a)).

Let L1 and L2 are two lower triangular n×n matrices and to prove that the product of two lower…

Answered: a) Let L₁ and L2 be two lower…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Question

**Properties of Triangular Matrices**

1. **Lower Triangular Matrices Multiplication:**
- **Statement:** Let $ L_1 $ and $ L_2 $ be two lower triangular $ n \times n $ matrices. Show that $ L_1L_2 $ is also lower triangular.
- **Hint:** The $(i, j)$-th entry of $ L_1L_2 $ is the inner product of the $ i $-th row of $ L_1 $ with the $ j $-th column of $ L_2 $. If $ i < j $, that is, if the entry we compute is above the diagonal, show that the inner product in question is 0.

2. **Upper Triangular Matrices Multiplication:**
- **Conclusion:** The product of two upper triangular $ n \times n $ matrices is also upper triangular (e.g., by using transposition, and part (a)).

Expert Solution

Step 1

Let $L_{1}$ and $L_{2}$ are two lower triangular $n \times n$ matrices and to prove that the product of two lower triangular matrices $L_{1} L_{2}$ is also lower triangular proceed as follows.

Let $(i, j)$ entry of matrix $L_{1}$ is $a_{i j}$ while that of matrix $L_{2}$ is $b_{i j}$ .

Since both the matrices $L_{1}$ and $L_{2}$ are both lower triangular matrices, therefore all the entries above the diagonal in $L_{1}, L_{2}$ are zero.

Any entry $a_{i j}$ in the matrix is above the above the diagonal if $i < j$ .

Therefore, $a_{i j} = 0 a n d b_{i j} = 0 i f i < j$ .