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Let A be an (n × n) matrix. We apply the elementary transformation of type 1 to A: "add m times row j to row i" (i.e. row i + mx row j), where m is a non-zero constant and j < i. a) Show that this transformation is encoded as a matrix multiplication L · A where L has the following form: 1 1 m 1 1 where m is in the entry (i, j). b) Write down a closed form for the matrix L-'. Multiplying with L-1 from the left also corresponds to an elementary transformation of type 1, what is this transformation?

Let A be an (n × n) matrix. We apply the elementary transformation of type 1 to A: "add m times row j to row i" (i.e. row i + mx row j), where m is a non-zero constant and j < i. a) Show that this transformation is encoded as a matrix multiplication L · A where L has the following form: 1 1 m 1 1 where m is in the entry (i, j). b) Write down a closed form for the matrix L-'. Multiplying with L-1 from the left also corresponds to an elementary transformation of type 1, what is this transformation?

Advanced Engineering Mathematics
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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Let A be an (n × n) matrix. We apply the elementary transformation of type 1 to A: "add m times row j to row i" (i.e. row i + mx row j), where m is a non-zero constant and j<i. a) Show that this transformation is encoded as a matrix multiplication L- A where L has the following form: 1 1 т 1 where m is in the entry (i, j). b) Write down a closed form for the matrix L-1. Multiplying with L-1 from the left also corresponds to an elementary transformation of type 1, what is this transformation?
Transcribed Image Text:Let A be an (n × n) matrix. We apply the elementary transformation of type 1 to A: "add m times row j to row i" (i.e. row i + mx row j), where m is a non-zero constant and j<i. a) Show that this transformation is encoded as a matrix multiplication L- A where L has the following form: 1 1 т 1 where m is in the entry (i, j). b) Write down a closed form for the matrix L-1. Multiplying with L-1 from the left also corresponds to an elementary transformation of type 1, what is this transformation?
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