Q89E Is the product of two lower tria... [FREE SOLUTION]

91影视

Linear Algebra With Applications

Otto Bretscher

$Math Studyset 91影视 Explanations$ Math

5th Edition 路 442 pages

ISBN: 9780321796974

Chapter 2: Q89E (page 102)

Is the product of two lower triangular matrices a lower triangular matrix as well? Explain your answer.

Short Answer

Expert verified

Yes, the product of two lower triangular matrices is a lower triangular matrix.

Step by step solution

Definition of a Lower triangular matrix

A lower triangular matrix is defined as a square matrix having all the entries above the main diagonal zero.

Consider two n×n lower triangular matrices

Let us consider two $n 脳 n$ lower triangular matricesx

$A = [\begin{matrix} a_{11} & 0 & 鈥� & 0 \\ a_{21} & a_{22} & 鈥� & 0 \\ 鈰� & 鈰� & 鈰� & 鈰� \\ a_{n 1} & a_{n 2} & 鈥� & a_{m m} \end{matrix}]$

and

$B = [\begin{matrix} b_{11} & 0 & 鈥� & 0 \\ b_{21} & b_{22} & 鈥� & 0 \\ 鈰� & 鈰� & 鈰� & 鈰� \\ b_{n 1} & b_{n 2} & 鈥� & b_{n n} \end{matrix}]$

Prove that the matrix AB is lower triangular matrices

We have to prove that the matrix AB is lower triangular, which means we have to show that the $i j^{t h}$ entry of AB is 0 whenever i < j.

Now, the $i j^{t h}$ entry of AB,localid="1659347524654" ${(A B)}_{i j} = (i^{t h} r o w o f A) : (j^{t h} c o l u m n s o f B)$ . Therefore

${(A B)}_{i j} = [a_{i 1} a_{i 2} . . . a_{i i} 0 . . . 0] [\begin{matrix} 0 \\ 鈰� \\ 0 \\ b_{j j} \\ 鈰� \\ b_{n j} \end{matrix}]$

which is equal to zero if i < j.

Therefore the product matrix AB is lower triangular.

The final answer

Yes, the product of two lower triangular matrices is a lower triangular matrix.

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91影视

Is the product of two lower triangular matrices a lower triangular matrix as well? Explain your answer.

Short Answer

Step by step solution

Definition of a Lower triangular matrix

Consider two n×n lower triangular matrices

Prove that the matrix AB is lower triangular matrices

The final answer

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