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2. True or False with explanation (e.g. a piece of the Invertible Matrix Theorem) or a counterexample. All matrices are n × n. (a) If the equation Ax = 0 has only the trivial solution, then A is row equivalent to the identity matrix. (b) If the columns of A span R", then they are linearly independent. (c) If AT is not invertible, then neither is A. (d) If there is a matrix D with AD = I, then DA = I also.

2. True or False with explanation (e.g. a piece of the Invertible Matrix Theorem) or a counterexample. All matrices are n × n. (a) If the equation Ax = 0 has only the trivial solution, then A is row equivalent to the identity matrix. (b) If the columns of A span R", then they are linearly independent. (c) If AT is not invertible, then neither is A. (d) If there is a matrix D with AD = I, then DA = I also.

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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2. True or False with explanation (e.g., a piece of the Invertible Matrix Theorem) or a counterexample. All matrices are \( n \times n \). (a) If the equation \( A\mathbf{x} = \mathbf{0} \) has only the trivial solution, then \( A \) is row equivalent to the identity matrix. (b) If the columns of \( A \) span \( \mathbb{R}^n \), then they are linearly independent. (c) If \( A^T \) is not invertible, then neither is \( A \). (d) If there is a matrix \( D \) with \( AD = I \), then \( DA = I \) also.
Transcribed Image Text:2. True or False with explanation (e.g., a piece of the Invertible Matrix Theorem) or a counterexample. All matrices are \( n \times n \). (a) If the equation \( A\mathbf{x} = \mathbf{0} \) has only the trivial solution, then \( A \) is row equivalent to the identity matrix. (b) If the columns of \( A \) span \( \mathbb{R}^n \), then they are linearly independent. (c) If \( A^T \) is not invertible, then neither is \( A \). (d) If there is a matrix \( D \) with \( AD = I \), then \( DA = I \) also.
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