Let \(A\) be a \(4 \times 4\) matrix with determinant 7 . Give a proof or counterexample for each of the following.

a) For some vector \(\mathbf{b}\) the equation \(A \mathbf{x}=\mathbf{b}\) has exactly one solution.

b) For some vector \(\mathbf{b}\) the equation \(A \mathbf{x}=\mathbf{b}\) has infinitely many solutions.

c) For some vector \(\mathbf{b}\) the equation \(A \mathbf{x}=\mathbf{b}\) has no solution.

d) For all vectors \(\mathbf{b}\) the equation \(A \mathbf{x}=\mathbf{b}\) has at least one solution.