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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.

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

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