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1 Methods Linear systems can generally be written in the form Ax = b (1) where A is an m × n matrix, x is a vector of length n, and b is a vector of length m. There can be no solution, one solution, or infinitely many solutions for such configuration. In matrix form this can be written as   a1,1 a1,2 · · · a1,n−1 a1,n a2,1 a2,2 · · · a2,n−1 a2,n . . . . . . . . . . . . . . . am−1,1 am−1,2 · · · am−1,n−1 am−1,n am,1 am,2 · · · am,n−1 am,n     x1 x2 . . . xn−1 xn   =   b1 b2 . . . bm−1 bm   (2) which corresponds to the linear system a1,1x1 + a1,2x2 + · · · + a1,n−1xn−1 + a1,nxn = b1 a2,1x1 + a2,2x2 + · · · + a2,n−1xn−1 + a2,nxn = b2 . . . (3) am−1,1x1 + am−1,2x2 + · · · + am−1,n−1xn−1 + am−1,nxn = bm−1 am,1x1 + am,2x2 + · · · + am,n−1xn−1 + am,nxn = bm 1.1 Gaussian Elimination For simplicity, let A be a square matrix. The premise of Gaussian elimination is to determine from the augmented matrix   a1,1 a1,2 · · · a1,n−1 a1,n b1 a2,1 a2,2 · · · a2,n−1 a2,n b2 . . . . . . . . . . . . . . . . . . an−1,1 an−1,2 · · · an−1,n−1 an−1,n bn−1 an,1 an,2 · · · an,n−1 an,n bn   (4) a row echelon form that can easily give the solutions of the system of equations. This is shown by   a 0 1,1 a 0 1,2 · · · a 0 1,n−1 a 0 1,n b 0 1 0 a 0 2,2 · · · a 0 2,n−1 a 0 2,n b 0 2 . . . . . . . . . . . . . . . . . . 0 0 · · · a 0 n−1,n−1 a 0 n−1,n b 0 n−1 0 0 · · · 0 a 0 n,n b 0 n   (5) and can be calculated by following the rules of solving linear equations: • a nonzero constant may be multiplied on both sides of any of the equation, • two rows can be switched with their corresponding b values, and • adding and/or subtracting two rows are allowed and placing it on one of those rows.

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