Skip to Main Content
It looks like you're using Internet Explorer 11 or older. This website works best with modern browsers such as the latest versions of Chrome, Firefox, Safari, and Edge. If you continue with this browser, you may see unexpected results.

# Linear Algebra

## Matrix Representation of a System of Linear Equations

You may noticed from the example in the previous page that the variables $$x_1, x_2, \ldots$$ were not moved or changed throughout the process. As long as we keep the coefficients lined up, we can write the system out as a rectangular array called a matrix.

 A general linear system of m equations in n unknowns can be represented by the matrix  $\left[ \begin{array}{cccccc|c} a_11 & a_12 & \ldots & a_1j & \ldots & a_1n & b_1 \\ a_21 & a_22 & \ldots & a_2j & \ldots & a_2n & b_2 \\ \vdots & \vdots & & \vdots & & \vdots & \vdots \\ a_i1 & a_i2 & \ldots & a_ij & \ldots & a_in & b_i \\ vdots & \vdots & & \vdots & & \vdots & \vdots \\ a_m1 & a_m2 & \ldots & a_mj & \ldots & a_mn & b_m \end{array} \right]$

For convenience, we denote the augmented matrix of a system with coefficient matrix A and right-hand side $$\vec{b} = \begin{bmatrix} b_1 \\ \vdots \\ b_m \end{bmatrix}$$ by $$\left[ \begin{array}{c|c} A & \vec{b} \end{array} \right]$$

Example: Write the coefficient matrix and augmented matrix for the following system:

\begin{array}{cccccc} 3s &+8t &-18u &+ v & = & 35 \\ s &+ 2t &-4u & &= & 11 \\ s &+ 3t &- 7u &+ v & = & 10 \end{array}

Solution:

$A = \left[\begin{array}{cccc} 3 &8 &-18 & 1 \\ 1 & 2 &-4 & 0 \\ 1 & 3 &- 7 & 1 \end{array} \right]$

$\left[ \begin{array}{cccc|c} 3 &8 &-18 & 1 & 35\\ 1 & 2 &-4 & 0 &11 \\ 1 & 3 &- 7 & 1 &10\end{array} \right]$

## Row Operations Explained

In Gaussian elimination, there are three types of elementary row operations:

• Swapping two rows,
• Multiplying a row by a non-zero number,
• Adding a multiple of one row to another row.

The objective of using a row operation is to get the augmented matrix into a triangular matrix.

Below are videos explaining how and why to perform each row operation during Gaussian elimination.

Designed by Matthew Cheung. This work is licensed under a Creative Commons Attribution 4.0 International License.
chat loading...