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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
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 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
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
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