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.

Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter9: Systems Of Equations And Inequalities
Section9.9: Properties Of Determinants
Problem 24E
Question
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.
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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