For this exercise assume that all matrices are ntimesn. 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 ntimesn matrix D such that AD equalsI, then there is also an ntimesn matrix C such that CA equalsI. Choose the correct answer below.
A. 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
B. True: by the Invertible Matrix Theorem if equation Ax= 0 has only the trivial solution, then the equation matrix is not invertible. Thus, A cannot be row equivalent to the nxn identity matrix. Ax - b has no solutions for each b in R". Thus, A must also be row equivalent to the n x n identity matrix °
C. 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 n x n identity matrix.
D. 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 R". Thus, A must also be row equivalent to the nx n identity matrix