Gauss-Jordan elimination

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A method to solve LaTeX: Ax=b that performs elementary row and column operations on LaTeX: A to annihilate successive elements of LaTeX: A in order to reduce LaTeX: A to an identity matrix. On paper, the same operations are applied to LaTeX: b, then the solution is obtained by solving the resulting identity system. In a computer, the matrices effecting the elementary operations are saved as elementary matrices, say LaTeX: E_i for the i-th operation. Then, the system is equivalent to LaTeX: E_1 E_2 \ldots E_n x = b, and forward transformation is applied to solve for LaTeX: x. This is what is done in the (revised) simplex method, and each iteration is a pivot operation.

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