以下題目為去年第一次期中考題，歡迎大家踴躍討論及解答：

Suppose that $A\in M_{m\times n}$ and $B\in M_{n\times m}$ such that $BA=I_n$.

(a) Prove that if $\mathbf{c}\in\mathbb{R}^m$ is a solution of the homogeneous system $(AB-I_m)\mathbf{x}=\mathbf{0}$, then the linear system $A\mathbf{x}=\mathbf{c}$ is consistent.

(b) Prove that if the linear system $A\mathbf{x}=\mathbf{c}$ is consistent, then $\mathbf{c}\in\mathbb{R}^m$ is a solution of the homogeneous system $(AB-I_m)\mathbf{x}=\mathbf{0}$.