# 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.