Gaussian Elimination

A method for solving Matrix Equations of the form

$${\hbox{\sf A}}{\bf x}={\bf b}.$$ (1)

Starting with the system of equations

$$\left[{\matrix{ a_{11} & a_{12} & \cdots & a_{1k}\cr a_{21... ...ht] = \left[{\matrix{b_1\cr b_2\cr \vdots\cr b_k\cr}}\right],$$ (2)

compose the augmented Matrix ``equation''

$$\left[\matrix{ a_{11} & a_{12} & \cdots & a_{1k}\cr a_{21}... ...right] \left[{\matrix{x_1\cr x_2\cr \vdots\cr x_k\cr}}\right].$$ (3)

Then, perform Matrix operations to put the augmented Matrix into the form

$$\left[\matrix{ a_{11}' & a_{12}' & \cdots & a_{1k}'\cr 0 &... ...right] \left[{\matrix{x_1\cr x_2\cr \vdots\cr x_k\cr}}\right].$$ (4)

Solve the equation of the $k$th row for $x_k$, then substitute back into the equation of the $(k-1)$st row obtain a solution for $x_{k-1}$, etc., according to the formula

$$x_i = {1\over a'_{ii}} \left({b'_i-\sum_{j=i+1}^k a'_{ij}x_j}\right).$$ (5)

See also Gauss-Jordan Elimination, LU Decomposition, Matrix Equation, Square Root Method