Chapter 2 - Matrices and Systems of Linear Equations - 2.6 The Inverse of a Square Matrix - Problems - Page 178: 42

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Let $A$ be an $n × n$ matrix Given: $Ax_i=b_i$ Multiply both sides by $A^{-1}$: $A^{-1}Ax_i=A^{-1}b_i$ Since $A^{-1}A=I$ then $Ix_i=A^{-1}b_i\\\rightarrow x_i=A^{-1}b_i$ Using Gauss-Jordan method, we write: $[A|b]$ as augmented matrix. Apply Gauss- Jordan elimination, we get: $[I|x_i]$ Hence, $I$ is a identity matrix and $x_i$ are its solutions.