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.
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 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.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F101db870-ee2e-4aa4-9595-a6c70c5a50a9%2F28c7f6ef-d527-4eae-a3b4-40bc687982ac%2Fen7qhmo_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 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.
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