6. Suppose that $A$ is an $n \times n$ matrix with $n \geq 6$. For an $n \times 1$ vector $b$ consider the equations $$\begin{aligned}
& A x = b , \text { for an } n \times 1 \text { vector } x \\
& A ^ { T } x = b \text { for an } n \times 1 \text { vector } x
\end{aligned}$$ where $A ^ { T }$ is the transpose of the matrix $A$. Which of the following statements are correct? (a) If equation (??) admits a solution for all $b$, then $A ^ { - 1 }$ exists. (b) If equation (??) admits a solution for all $b$, then equation (??) also admits a solution for all $b$. (c) If equation (??) admits a solution for some $b$, then $A ^ { - 1 }$ exists. (d) If equation (??) admits a solution for some $b$, then equation (??) also admits a solution for that $b$.
6. Suppose that $A$ is an $n \times n$ matrix with $n \geq 6$. For an $n \times 1$ vector $b$ consider the equations
$$\begin{aligned}
& A x = b , \text { for an } n \times 1 \text { vector } x \\
& A ^ { T } x = b \text { for an } n \times 1 \text { vector } x
\end{aligned}$$
where $A ^ { T }$ is the transpose of the matrix $A$. Which of the following statements are correct?\\
(a) If equation (??) admits a solution for all $b$, then $A ^ { - 1 }$ exists.\\
(b) If equation (??) admits a solution for all $b$, then equation (??) also admits a solution for all $b$.\\
(c) If equation (??) admits a solution for some $b$, then $A ^ { - 1 }$ exists.\\
(d) If equation (??) admits a solution for some $b$, then equation (??) also admits a solution for that $b$.\\