Let A be a lower triangular nxn matrix with nonzero entries on the diagonal. Show that A is invertible and A 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 also transforms I into the identity matrix A inverse the zero matrix the zero matrix. A. the inverse of A.

Let A be a lower triangular nxn matrix with nonzero entries on the diagonal. Show that A is invertible and A 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 also transforms I into the identity matrix A inverse the zero matrix the zero matrix. A. the inverse of A.

Linear Algebra: A Modern Introduction

4th Edition

ISBN:9781285463247

Author:David Poole

Publisher:David Poole

Chapter5: Orthogonality

Section5.3: The Gram-schmidt Process And The Qr Factorization

Problem 23EQ

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