xplain why A can be changed into I using only row replacements and scaling. (Where are the pivots?) Also, explain why the row perations that reduce A to I change I into a lower triangular matrix.] consider the augmented matrix [A I]. Remember, an nxn matrix A is invertible if and only if A is row equivalent to I, and in this case, any equence of elementary row operations that reduces A to the identity matrix also transforms I, into the inverse of A. he (1.)-entry can be scaled to and the entries below it can be changed to by This affects only the column of A and the column of I. So the 2.-entry in the new matrix is still nonzero and is now the only nonzero entry of row 2 in the first n columns because A was lower angular). Type whole numbers.) multiplying the rows below by multiniving the nth second first third
xplain why A can be changed into I using only row replacements and scaling. (Where are the pivots?) Also, explain why the row perations that reduce A to I change I into a lower triangular matrix.] consider the augmented matrix [A I]. Remember, an nxn matrix A is invertible if and only if A is row equivalent to I, and in this case, any equence of elementary row operations that reduces A to the identity matrix also transforms I, into the inverse of A. he (1.)-entry can be scaled to and the entries below it can be changed to by This affects only the column of A and the column of I. So the 2.-entry in the new matrix is still nonzero and is now the only nonzero entry of row 2 in the first n columns because A was lower angular). Type whole numbers.) multiplying the rows below by multiniving the nth second first third
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
![Let A be a lower triangular nxn matrix with nonzero entries on the diagonal. Show that A is invertible and A¹ is lower triangular. [Hint:
Explain why A can be changed into I using only row replacements and scaling. (Where are the pivots?) Also, explain why the row
operations that reduce A to I change I into a lower triangular matrix.]
Consider the augmented matrix [A. I]. Remember, an nxn matrix A is invertible if and only if A is row equivalent to I, and in this case, any
sequence of elementary row operations that reduces A to the identity matrix also transforms I into the inverse of A.
The (1. ])-entry can be scaled to and the entries below it can be changed to by
This affects only the
column of A and the
column of I. So the
(2, -entry in the new matrix is still nonzero and is now the only nonzero entry of row 2 in the first n columns because A was lower
triangular).
(Type whole numbers.)
multiplying the rows below by
multiplying the next row below by
adding multiples of row 1 to the rows below.
adding multiples of row 1 to the next row below.
nth
second
first
third
first
third
second
nth](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F101db870-ee2e-4aa4-9595-a6c70c5a50a9%2F8d5c8d15-2023-4030-88c7-3a4acbe08dba%2Fycy6c56_processed.png&w=3840&q=75)
Transcribed Image Text:Let A be a lower triangular nxn matrix with nonzero entries on the diagonal. Show that A is invertible and A¹ is lower triangular. [Hint:
Explain why A can be changed into I using only row replacements and scaling. (Where are the pivots?) Also, explain why the row
operations that reduce A to I change I into a lower triangular matrix.]
Consider the augmented matrix [A. I]. Remember, an nxn matrix A is invertible if and only if A is row equivalent to I, and in this case, any
sequence of elementary row operations that reduces A to the identity matrix also transforms I into the inverse of A.
The (1. ])-entry can be scaled to and the entries below it can be changed to by
This affects only the
column of A and the
column of I. So the
(2, -entry in the new matrix is still nonzero and is now the only nonzero entry of row 2 in the first n columns because A was lower
triangular).
(Type whole numbers.)
multiplying the rows below by
multiplying the next row below by
adding multiples of row 1 to the rows below.
adding multiples of row 1 to the next row below.
nth
second
first
third
first
third
second
nth
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
Recommended textbooks for you
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
