[6] [Inverses and products of lower triangular matrices] (a) Suppose L is a nonsingular 4 × 4 lower triangular matrix: Show that if L: = r = 0 121 122 131 41 0 0 O 132 133 142 143 144/ 711 Y21 0 0 712 113 114 22 23 24 31 32 33 34 741 42 43 44/ is such that LI = Id where Id is the 4 × 4 identity matrix, then I must also be lower triangular. (Note: this shows that the 'right-inverse' of L must be lower triangular; a nearly identical computation shows its 'left-inverse' also must be lower triangular, which then implies L-¹ must be lower triangular.) (b) Suppose that L₁ and L₂ are both 4 × 4 lower triangular matrices. Show that their product L₁ L2 is also lower triangular.
[6] [Inverses and products of lower triangular matrices] (a) Suppose L is a nonsingular 4 × 4 lower triangular matrix: Show that if L: = r = 0 121 122 131 41 0 0 O 132 133 142 143 144/ 711 Y21 0 0 712 113 114 22 23 24 31 32 33 34 741 42 43 44/ is such that LI = Id where Id is the 4 × 4 identity matrix, then I must also be lower triangular. (Note: this shows that the 'right-inverse' of L must be lower triangular; a nearly identical computation shows its 'left-inverse' also must be lower triangular, which then implies L-¹ must be lower triangular.) (b) Suppose that L₁ and L₂ are both 4 × 4 lower triangular matrices. Show that their product L₁ L2 is also lower triangular.
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter4: Eigenvalues And Eigenvectors
Section4.4: Similarity And Diagonalization
Problem 40EQ
Related questions
Question
Lower triangular matrices
![[6] [Inverses and products of lower triangular matrices]
(a) Suppose L is a nonsingular 4 × 4 lower triangular matrix:
Show that if
L=
=
r =
111 0 0 0
121 122 0 0
131 132 133 0
141 142 143 144)
711 712 713 714
21
22 23 24
31
32
33
34
Y41 42 43
44,
is such that LI
be lower triangular.
(Note: this shows that the ‘right-inverse' of L must be lower triangular;
a nearly identical computation shows its 'left-inverse' also must be lower
triangular, which then implies L-¹ must be lower triangular.)
(b) Suppose that L₁ and L₂ are both 4 × 4 lower triangular matrices.
Show that their product L₁ L₂ is also lower triangular.
Id where Id is the 4 × 4 identity matrix, then I must also](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F7fd47556-f3ce-4f39-818d-be563d9523c8%2Ff5d2d4b0-e2f5-4a1a-bdf1-1f083f3b1356%2Fml63s8y_processed.png&w=3840&q=75)
Transcribed Image Text:[6] [Inverses and products of lower triangular matrices]
(a) Suppose L is a nonsingular 4 × 4 lower triangular matrix:
Show that if
L=
=
r =
111 0 0 0
121 122 0 0
131 132 133 0
141 142 143 144)
711 712 713 714
21
22 23 24
31
32
33
34
Y41 42 43
44,
is such that LI
be lower triangular.
(Note: this shows that the ‘right-inverse' of L must be lower triangular;
a nearly identical computation shows its 'left-inverse' also must be lower
triangular, which then implies L-¹ must be lower triangular.)
(b) Suppose that L₁ and L₂ are both 4 × 4 lower triangular matrices.
Show that their product L₁ L₂ is also lower triangular.
Id where Id is the 4 × 4 identity matrix, then I must also
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
This is a popular solution!
Trending now
This is a popular solution!
Step by step
Solved in 2 steps with 2 images
Recommended textbooks for you
Linear Algebra: A Modern Introduction
Algebra
ISBN:
9781285463247
Author:
David Poole
Publisher:
Cengage Learning
Elementary Linear Algebra (MindTap Course List)
Algebra
ISBN:
9781305658004
Author:
Ron Larson
Publisher:
Cengage Learning
Algebra for College Students
Algebra
ISBN:
9781285195780
Author:
Jerome E. Kaufmann, Karen L. Schwitters
Publisher:
Cengage Learning
Linear Algebra: A Modern Introduction
Algebra
ISBN:
9781285463247
Author:
David Poole
Publisher:
Cengage Learning
Elementary Linear Algebra (MindTap Course List)
Algebra
ISBN:
9781305658004
Author:
Ron Larson
Publisher:
Cengage Learning
Algebra for College Students
Algebra
ISBN:
9781285195780
Author:
Jerome E. Kaufmann, Karen L. Schwitters
Publisher:
Cengage Learning
College Algebra (MindTap Course List)
Algebra
ISBN:
9781305652231
Author:
R. David Gustafson, Jeff Hughes
Publisher:
Cengage Learning
Algebra and Trigonometry (MindTap Course List)
Algebra
ISBN:
9781305071742
Author:
James Stewart, Lothar Redlin, Saleem Watson
Publisher:
Cengage Learning
College Algebra
Algebra
ISBN:
9781305115545
Author:
James Stewart, Lothar Redlin, Saleem Watson
Publisher:
Cengage Learning