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Row reduction

Linear systems & matrices

Japanese school year: University year 1

What you learn

This section covers Gaussian and Gauss-Jordan elimination, solving systems of linear equations systematically via elementary row operations on augmented matrices. It provides an efficient algorithmic approach for handling large systems and computing inverses. Prior knowledge of elementary algebra and basic matrix notation is recommended.

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Key points

These are the three elementary row operations used to solve linear systems with matrices. Multiplying a row by a non-zero number, swapping two rows, or adding a multiple of one row to another will never change the solutions.

This describes the row reduction process (Gaussian elimination) for solving linear systems. Clear the numbers below each pivot to form a staircase shape, and then clear the numbers above to read off solutions directly.

This procedure finds the inverse matrix A−1A^{-1} using row operations. Place the identity matrix II next to matrix AA; once row reduction turns the left side into II, the right side becomes A−1A^{-1}.

[A∣I]→[I∣A−1][A \mid I] \to [I \mid A^{-1}]

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