# Gaussian Elimination

### Row operations:

 0 1 0 1 0 0 0 0 1
Interchange rows       and

 2 0 0 0 1 0 0 0 1
Multiply row       by    /

 1 0 0 2 1 0 0 0 1
Add    /    times row       to row

### Row Operations Performed:

Gaussian elimination is a method for solving a system of linear equations. The procedure can be used for any number of equations in any number of variables. Given a system of $$n$$ equations in $$m$$ variables \begin{align*} a_{11}x_1 + a_{12}x_2 + &\cdots + a_{1m}x_m = y_1 \\ a_{21}x_1 + a_{22}x_2 + &\cdots + a_{2m}x_m = y_2 \\ &\vdots \\ a_{n1}x_1 + a_{n2}x_2 + &\cdots + a_{nm}x_m = y_n \end{align*} we can create the augmented matrix $\left[\begin{array}{cccc|c} a_{11} & a_{12} & \cdots & a_{1m} & y_1 \\ a_{21} & a_{22} & \cdots & a_{2m} & y_2 \\ \vdots&\vdots& \ddots & \vdots & \vdots \\ a_{n1} & a_{n2} & \cdots & a_{nm} & y_n \\ \end{array}\right]$ Elementary row operations are performed on this system to reduce it to row echelon form. If the matrix with entries $$(a_{ij})$$ is an invertible, square matrix, then the left side of the augmented matrix can be reduced to the identity and the system has a unique solution. Otherwise, the system may have multiple or no solutions. The reduced form of the matrix is independent of the operations chosen.