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Let \(\mathbf{A}\) be an invertible matrix. Suppose there are two matrices, \(\mathbf{B}\) and \(\mathbf{C}\), such that \(\mathbf{A}\mathbf{B} = \mathbf{B}\mathbf{A} = \mathbf{I}\) and \(\mathbf{A}\mathbf{C} = \mathbf{C}\mathbf{A} = \mathbf{I}\). Prove that \(\mathbf{B}=\mathbf{C}\). (This shows the inverse is unique).
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