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Prerequisites: Basic vector and matrix operations. Elementary Row Operations and Reduced Form There are three types of elementary row operations: Scaling: Multip a row by a scalar $k$ Addition: Add $k$ times one row to another row Swapping: Exchange two rows Each operation corresponds to multiplying...

Gaussian elimination solves systems of linear equations in $O(n^3)$ time. Consider a system of $n$ linear equations in $n$ unknowns: $$ a_{11}x_1 + a_{12}x_2 + \cdots + a_{1n}x_n = b_1 $$ $$ \vdots $$ $$ a_{n1}x_1 + a_{n2}x_2 + \cdots + a_{nn}x_n = b_n $$ Gauss-Jordan Elimination Algorithm: Select a...