For each of the following matrices $A$ : (a) Determine the rank and the dimensions of the four fundamental subspaces. (b) Find bases for both the kernel and cokernel. (c) Find explicit conditions on vectors $\mathbf{b}$ that guarantee that the system $A \mathbf{x}=\mathbf{b}$ has a solution. (d) Write down a specific nonzero vector $\mathbf{b}$ that satisfies your conditions, and then find all possible solutions $\mathbf{x}$.
(i) $\left(\begin{array}{rr}1 & 2 \\ -2 & -4\end{array}\right)$,
(ii) $\left(\begin{array}{rrr}3 & -1 & -2 \\ -6 & 2 & 4\end{array}\right)$,
(iii) $\left(\begin{array}{rr}1 & 5 \\ -2 & 3 \\ 2 & 7\end{array}\right)$,
(iv) $\left(\begin{array}{rrr}2 & -5 & -1 \\ 1 & -6 & -4 \\ 3 & -4 & 2\end{array}\right)$,
(v) $\left(\begin{array}{rrr}2 & 5 & 7 \\ 6 & 13 & 19 \\ 3 & 8 & 11 \\ 1 & 2 & 3\end{array}\right)$,
(vi) $\left(\begin{array}{rrrr}1 & 2 & 3 & 4 \\ 3 & 2 & 4 & 1 \\ 1 & -2 & 2 & 7 \\ 3 & 6 & 5 & -2\end{array}\right)$,
(vii)
$\left(\begin{array}{rrrrr}2 & 4 & 0 & -6 & 0 \\ 1 & 2 & 3 & 15 & 0 \\ 3 & 6 & -1 & 15 & 5 \\ -3 & -6 & 2 & 21 & -6\end{array}\right)$