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is an invertible matrix. A is row equivalent to the n x n identity matrix. A has n pivot positions. The equation Ax=0 has only the trivial solution. The columns of A form a linearly independent set. The linear transformation x Ax is one-to-one. The equation Ax=b has at least one solution for each b € Rn. The columns of A span R". The linear transformation x→ Ax maps R" onto R". There is an n x n matrix C such that CA = I. There is an n x n matrix D such that AD = I. AT is an invertible matrix.

is an invertible matrix. A is row equivalent to the n x n identity matrix. A has n pivot positions. The equation Ax=0 has only the trivial solution. The columns of A form a linearly independent set. The linear transformation x Ax is one-to-one. The equation Ax=b has at least one solution for each b € Rn. The columns of A span R". The linear transformation x→ Ax maps R" onto R". There is an n x n matrix C such that CA = I. There is an n x n matrix D such that AD = I. AT is an invertible matrix.

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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Theorem 3 Let A be a square n × n matrix. Then the following statements are equivalent. That is, for a
given A, the statements are either all true or all false. state true or false and indicate why.

A is an invertible matrix. A is row equivalent to the n x n identity matrix. A has n pivot positions. The equation Ax = 0 has only the trivial solution. The columns of A form a linearly independent set. The linear transformation x→ Ax is one-to-one. The equation Ax=b has at least one solution for each b E R². The columns of A span Rn. The linear transformation x Ax maps R" onto R". There is an n x n matriz C such that CA = I. There is an n x n matrix D such that AD = I. AT is an invertible matrix.
Transcribed Image Text:A is an invertible matrix. A is row equivalent to the n x n identity matrix. A has n pivot positions. The equation Ax = 0 has only the trivial solution. The columns of A form a linearly independent set. The linear transformation x→ Ax is one-to-one. The equation Ax=b has at least one solution for each b E R². The columns of A span Rn. The linear transformation x Ax maps R" onto R". There is an n x n matriz C such that CA = I. There is an n x n matrix D such that AD = I. AT is an invertible matrix.
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