Question

For this exercise assume that all matrices are n × n. Each part of 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. An implication is False if there is an instance in which "statement 2" is false but "statement 1" is true. Complete parts (a) through (e). Justify each answer. a. If there is an n × n matrix D such that AD = I, then there is also an n × n matrix C such that CA = I. Choose the correct answer below. A. True; by the Invertible Matrix Theorem, if there is an n × n matrix D such that AD = I, then it must be true that there is also an n × n matrix C such that CA = I. B. True; by the Invertible Matrix Theorem if AD = I then A, D, or both is/are the identity matrix. Therefore, CA = I. C. False; it does not follow from the Invertible Matrix Theorem that if AD = I, then CA = I. D. False, because matrix multiplication is not commutative. It is possible that only D (and not A) is invertible in the equation AD = I. This implies that CA = I is only true when C is invertible (for cases where A is not invertible), but it is not given that C is invertible. b. If the columns of A are linearly independent, then the columns of A span ??. Choose the correct answer below. A. True; by the Invertible Matrix Theorem if the columns of A are linearly dependent, then the columns of A must span ??. B. True; by the Invertible Matrix Theorem if the columns of A are linearly independent, then the columns of A must span ??. C. False; by the Invertible Matrix Theorem if the columns of A are linearly independent, then the columns of A do not span ??. D. False; by the Invertible Matrix Theorem if the columns of A are linearly dependent, then the columns of A must span ??. c. If the equation Ax = b has at least one solution for each b in ??, then the solution is unique for each b. A. True, but only for x ? 0; by the Invertible Matrix Theorem if Ax = b has at least one solution for each b in ??, then the equation Ax = 0 does not only have the trivial solution. B. False; by the Invertible Matrix Theorem if Ax = b has at least one solution for each b in ??, then the linear transformation x ? Ax does not map ?? onto ??. C. False; by the Invertible Matrix Theorem if Ax = b has at least one solution for each b in ??, then matrix A is not invertible. If A is not invertible, then according to the invertible matrix theorem the solution is not unique for each b. D. True; by the Invertible Matrix Theorem if Ax = b has at least one solution for each b in ??, then matrix A is invertible. If A is invertible, then according to the invertible matrix theorem the solution is unique for each b. d. If the linear transformation (x) ? Ax maps ?? into ??, then A has n pivot positions. A. True; the linear transformation (x) ? Ax will always map ?? into ?? for any n × n matrix. Therefore, according to the Invertible Matrix Theorem A has n pivot positions. B. False; according to the Invertible Matrix Theorem, (x) ? Ax maps ?? into ??, then A has n + 2 pivot positions. C. True; according to the Invertible Matrix Theorem if (x) ? Ax maps ?? into ?? then A is invertible, and if a matrix is invertible it has n pivot positions. D. False; the linear transformation (x) ? Ax will always map ?? into ?? for any n × n matrix. According to the Invertible Matrix Theorem A has n pivot positions only if (x) ? Ax maps ?? onto ??. e. If there is a b in ?? such that the equation Ax = b is inconsistent, then the transformation x ? Ax is not one-to-one. A. True; according to the Invertible Matrix Theorem if there is a b in ?? such that the equation Ax = b is inconsistent, then matrix A is invertible. B. True; according to the Invertible Matrix Theorem if there is a b in ?? such that the equation Ax = b is inconsistent, then equation Ax = b does not have at least one solution for each b in ?? and this makes A not invertible. C. False; according to the Invertible Matrix Theorem if there is a b in ?? such that the equation Ax = b is inconsistent, then equation Ax = b has at least one solution for each b in ?? and this makes A invertible. D. False; according to the Invertible Matrix Theorem if there is a b in ?? such that the equation Ax = b is inconsistent, then the linear transformation x ? Ax maps ?? onto ??.

          For this exercise assume that all matrices are n × n. Each part of 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. An implication is False if there is an instance in which "statement 2" is false but "statement 1" is true. Complete parts (a) through (e). Justify each answer.
a. If there is an n × n matrix D such that AD = I, then there is also an n × n matrix C such that CA = I. Choose the correct answer below.
A. True; by the Invertible Matrix Theorem, if there is an n × n matrix D such that AD = I, then it must be true that there is also an n × n matrix C such that CA = I.
B. True; by the Invertible Matrix Theorem if AD = I then A, D, or both is/are the identity matrix. Therefore, CA = I.
C. False; it does not follow from the Invertible Matrix Theorem that if AD = I, then CA = I.
D. False, because matrix multiplication is not commutative. It is possible that only D (and not A) is invertible in the equation AD = I. This implies that CA = I is only true when C is invertible (for cases where A is not invertible), but it is not given that C is invertible.
b. If the columns of A are linearly independent, then the columns of A span ??. Choose the correct answer below.
A. True; by the Invertible Matrix Theorem if the columns of A are linearly dependent, then the columns of A must span ??.
B. True; by the Invertible Matrix Theorem if the columns of A are linearly independent, then the columns of A must span ??.
C. False; by the Invertible Matrix Theorem if the columns of A are linearly independent, then the columns of A do not span ??.
D. False; by the Invertible Matrix Theorem if the columns of A are linearly dependent, then the columns of A must span ??.
c. If the equation Ax = b has at least one solution for each b in ??, then the solution is unique for each b.
A. True, but only for x ? 0; by the Invertible Matrix Theorem if Ax = b has at least one solution for each b in ??, then the equation Ax = 0 does not only have the trivial solution.
B. False; by the Invertible Matrix Theorem if Ax = b has at least one solution for each b in ??, then the linear transformation x ? Ax does not map ?? onto ??.
C. False; by the Invertible Matrix Theorem if Ax = b has at least one solution for each b in ??, then matrix A is not invertible. If A is not invertible, then according to the invertible matrix theorem the solution is not unique for each b.
D. True; by the Invertible Matrix Theorem if Ax = b has at least one solution for each b in ??, then matrix A is invertible. If A is invertible, then according to the invertible matrix theorem the solution is unique for each b.
d. If the linear transformation (x) ? Ax maps ?? into ??, then A has n pivot positions.
A. True; the linear transformation (x) ? Ax will always map ?? into ?? for any n × n matrix. Therefore, according to the Invertible Matrix Theorem A has n pivot positions.
B. False; according to the Invertible Matrix Theorem, (x) ? Ax maps ?? into ??, then A has n + 2 pivot positions.
C. True; according to the Invertible Matrix Theorem if (x) ? Ax maps ?? into ?? then A is invertible, and if a matrix is invertible it has n pivot positions.
D. False; the linear transformation (x) ? Ax will always map ?? into ?? for any n × n matrix. According to the Invertible Matrix Theorem A has n pivot positions only if (x) ? Ax maps ?? onto ??.
e. If there is a b in ?? such that the equation Ax = b is inconsistent, then the transformation x ? Ax is not one-to-one.
A. True; according to the Invertible Matrix Theorem if there is a b in ?? such that the equation Ax = b is inconsistent, then matrix A is invertible.
B. True; according to the Invertible Matrix Theorem if there is a b in ?? such that the equation Ax = b is inconsistent, then equation Ax = b does not have at least one solution for each b in ?? and this makes A not invertible.
C. False; according to the Invertible Matrix Theorem if there is a b in ?? such that the equation Ax = b is inconsistent, then equation Ax = b has at least one solution for each b in ?? and this makes A invertible.
D. False; according to the Invertible Matrix Theorem if there is a b in ?? such that the equation Ax = b is inconsistent, then the linear transformation x ? Ax maps ?? onto ??.

For this exercise assume that all matrices are n × n. Each part of 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. An implication is False if there is an instance in which "statement 2" is false but "statement 1" is true. Complete parts (a) through (e). Justify each answer.
a. If there is an n × n matrix D such that AD = I, then there is also an n × n matrix C such that CA = I. Choose the correct answer below.
A. True; by the Invertible Matrix Theorem, if there is an n × n matrix D such that AD = I, then it must be true that there is also an n × n matrix C such that CA = I.
B. True; by the Invertible Matrix Theorem if AD = I then A, D, or both is/are the identity matrix. Therefore, CA = I.
C. False; it does not follow from the Invertible Matrix Theorem that if AD = I, then CA = I.
D. False, because matrix multiplication is not commutative. It is possible that only D (and not A) is invertible in the equation AD = I. This implies that CA = I is only true when C is invertible (for cases where A is not invertible), but it is not given that C is invertible.
b. If the columns of A are linearly independent, then the columns of A span ??. Choose the correct answer below.
A. True; by the Invertible Matrix Theorem if the columns of A are linearly dependent, then the columns of A must span ??.
B. True; by the Invertible Matrix Theorem if the columns of A are linearly independent, then the columns of A must span ??.
C. False; by the Invertible Matrix Theorem if the columns of A are linearly independent, then the columns of A do not span ??.
D. False; by the Invertible Matrix Theorem if the columns of A are linearly dependent, then the columns of A must span ??.
c. If the equation Ax = b has at least one solution for each b in ??, then the solution is unique for each b.
A. True, but only for x ? 0; by the Invertible Matrix Theorem if Ax = b has at least one solution for each b in ??, then the equation Ax = 0 does not only have the trivial solution.
B. False; by the Invertible Matrix Theorem if Ax = b has at least one solution for each b in ??, then the linear transformation x ? Ax does not map ?? onto ??.
C. False; by the Invertible Matrix Theorem if Ax = b has at least one solution for each b in ??, then matrix A is not invertible. If A is not invertible, then according to the invertible matrix theorem the solution is not unique for each b.
D. True; by the Invertible Matrix Theorem if Ax = b has at least one solution for each b in ??, then matrix A is invertible. If A is invertible, then according to the invertible matrix theorem the solution is unique for each b.
d. If the linear transformation (x) ? Ax maps ?? into ??, then A has n pivot positions.
A. True; the linear transformation (x) ? Ax will always map ?? into ?? for any n × n matrix. Therefore, according to the Invertible Matrix Theorem A has n pivot positions.
B. False; according to the Invertible Matrix Theorem, (x) ? Ax maps ?? into ??, then A has n + 2 pivot positions.
C. True; according to the Invertible Matrix Theorem if (x) ? Ax maps ?? into ?? then A is invertible, and if a matrix is invertible it has n pivot positions.
D. False; the linear transformation (x) ? Ax will always map ?? into ?? for any n × n matrix. According to the Invertible Matrix Theorem A has n pivot positions only if (x) ? Ax maps ?? onto ??.
e. If there is a b in ?? such that the equation Ax = b is inconsistent, then the transformation x ? Ax is not one-to-one.
A. True; according to the Invertible Matrix Theorem if there is a b in ?? such that the equation Ax = b is inconsistent, then matrix A is invertible.
B. True; according to the Invertible Matrix Theorem if there is a b in ?? such that the equation Ax = b is inconsistent, then equation Ax = b does not have at least one solution for each b in ?? and this makes A not invertible.
C. False; according to the Invertible Matrix Theorem if there is a b in ?? such that the equation Ax = b is inconsistent, then equation Ax = b has at least one solution for each b in ?? and this makes A invertible.
D. False; according to the Invertible Matrix Theorem if there is a b in ?? such that the equation Ax = b is inconsistent, then the linear transformation x ? Ax maps ?? onto ??.

Added by Rachael H.

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