(Solved): For this exercise assume that the matrices are all nn. The statement in this exercise is an impli ...

For this exercise assume that the matrices are all $n\times n$. The statement in this exercise is an implication of the form "if "statement 1", then "statement $2^{\text{" }}$. Mark an implication as True if the truth of "statement $2^{\text{" }}$ 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 there is an $n\times n$ matrix $D$ such that $AD=I$, then there is also an $n\times n$ matrix $C$ such that $CA=1$. Choose the correct answer below. A. The statement is true. By the invertible Matrix Theorem, if $AD=I$ then $A,D$, or both is/are the identity matrix. Therefore, $\mathrm{CA}=1$. B. The statement is false. It does not follow from the invertble Matrix Theorem that if $AD=1$, then $CA=1$. C. The statement is true. By the Invertible Matrix Theorem, if there is an $n\times n$ matrix $D$ such that $AD=I$, then it must be true that there is also an $n\times n$ matrix $\mathrm{C}$ such that $\mathrm{CA}=1$. D. The statement is false. Matrix multiplication is not commutative. It is possible that only $D$ (and not A) is invertible in the equation $AD=1$. 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 $\mathrm{C}$ is invertible.