For this exercise assume that the matrices are all nxn. The statement in this exercise is an implication of the form "If "statement 1", then "statement 2"." Mark an implication as True if the truth of "statement 2" always follows whenever "statement 1" happens to be true. Mark the implication as False if "statement 2" is false but "statement 1" is true. Justify your answer. If the equation Ax = 0 has only the trivial solution, then A is row equivalent to the nxn identity matrix. O A. The statement is true. By the Invertible Matrix Theorem, if equation Ax = 0 has only the trivial solution, then the equation Ax = b has no solutions for each b in Rn. Thus, A must also be row equivalent to the nxn identity matrix. B. The statement is false. By the Invertible Matrix Theorem, if the equation Ax = 0 has only the trivial solution, then the matrix is not invertible; this means the columns of A do not span Rn. Thus, A must also be row equivalent to the nxn identity matrix. C. The statement is true. By the Invertible Matrix Theorem, if the equation Ax = 0 has only the trivial solution, then the matrix is invertible. Thus, A must also be row equivalent to the nxn identity matrix. D. The statement is false. By the Invertible Matrix Theorem, if the equation Ax = 0 has only the trivial solution, then the matrix is not invertible. Thus, A cannot be row equivalent to the nxn identity matrix.
For this exercise assume that the matrices are all nxn. The statement in this exercise is an implication of the form "If "statement 1", then "statement 2"." Mark an implication as True if the truth of "statement 2" always follows whenever "statement 1" happens to be true. Mark the implication as False if "statement 2" is false but "statement 1" is true. Justify your answer. If the equation Ax = 0 has only the trivial solution, then A is row equivalent to the nxn identity matrix. O A. The statement is true. By the Invertible Matrix Theorem, if equation Ax = 0 has only the trivial solution, then the equation Ax = b has no solutions for each b in Rn. Thus, A must also be row equivalent to the nxn identity matrix. B. The statement is false. By the Invertible Matrix Theorem, if the equation Ax = 0 has only the trivial solution, then the matrix is not invertible; this means the columns of A do not span Rn. Thus, A must also be row equivalent to the nxn identity matrix. C. The statement is true. By the Invertible Matrix Theorem, if the equation Ax = 0 has only the trivial solution, then the matrix is invertible. Thus, A must also be row equivalent to the nxn identity matrix. D. The statement is false. By the Invertible Matrix Theorem, if the equation Ax = 0 has only the trivial solution, then the matrix is not invertible. Thus, A cannot be row equivalent to the nxn identity matrix.
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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Question
Transcribed Image Text:For this exercise assume that the matrices are all nxn. The statement in
this exercise is an implication of the form "If "statement 1", then
"statement 2"." Mark an implication as True if the truth of "statement 2"
always follows whenever "statement 1" happens to be true. Mark the
implication as False if "statement 2" is false but "statement 1" is true.
Justify your answer.
If the equation Ax = 0 has only the trivial solution, then A is row
equivalent to the nxn identity matrix.
O A. The statement is true. By the Invertible Matrix Theorem, if
equation Ax = 0 has only the trivial solution, then the equation
Ax = b has no solutions for each b in Rn. Thus, A must also be
row equivalent to the nxn identity matrix.
B. The statement is false. By the Invertible Matrix Theorem, if the
equation Ax = 0 has only the trivial solution, then the matrix is not
invertible; this means the columns of A do not span Rn. Thus, A
must also be row equivalent to the nxn identity matrix.
C. The statement is true. By the Invertible Matrix Theorem, if the
equation Ax = 0 has only the trivial solution, then the matrix is
invertible. Thus, A must also be row equivalent to the nxn
identity matrix.
D. The statement is false. By the Invertible Matrix Theorem, if the
equation Ax = 0 has only the trivial solution, then the matrix is not
invertible. Thus, A cannot be row equivalent to the nxn identity
matrix.
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