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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.
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