Study Notes: Systems of Equations & Matrices (Gaussian Elimination)

Systems of Equations & Matrices

Gaussian Elimination

Gaussian elimination is a systematic method for solving systems of linear equations. It involves transforming the system's augmented matrix into row-echelon form using elementary row operations, making it easier to solve for the variables.

• System of Linear Equations: A set of equations where each equation is linear in the variables.

• Augmented Matrix: A matrix that includes the coefficients and constants from a system of equations.

• Row-Echelon Form: A form of a matrix where all nonzero rows are above any rows of all zeros, and each leading coefficient is to the right of the leading coefficient of the row above it.

Example System

Consider the system:

• 3x + y + 5z = 16

• 2x + y + 4z = 14

• x + y + z = 6

Step 1: Write the Augmented Matrix

The system can be written as an augmented matrix:

$\begin{bmatrix} 3 & 1 & 5 & | & 16 \\ 2 & 1 & 4 & | & 14 \\ 1 & 1 & 1 & | & 6 \end{bmatrix}$

Step 2: Row Reduction (Gaussian Elimination)

Use elementary row operations to convert the matrix to row-echelon form:

1. Make the leading coefficient of the first row 1 (if necessary, swap rows or divide).

2. Eliminate the x-term from the second and third rows by subtracting appropriate multiples of the first row.

3. Continue to the next column and repeat the process for y and z.

Example: Subtract 2 times the third row from the second row to eliminate x from the second equation.

Step 3: Back Substitution

Once the matrix is in row-echelon form, solve for the variables starting from the bottom row and substituting upwards.

Step 4: Interpreting the Solution

The solution to the system is the set of values for x, y, and z that satisfy all equations simultaneously.

Associated Matrix A

The coefficient matrix A for the system is:

$A = \begin{bmatrix} 3 & 1 & 5 \\ 2 & 1 & 4 \\ 1 & 1 & 1 \end{bmatrix}$

Summary Table: Steps in Gaussian Elimination

Step	Description
1	Write the augmented matrix for the system
2	Use row operations to reach row-echelon form
3	Back substitute to find variable values

Key Points

• Gaussian elimination is a standard method for solving systems of equations.

• Row operations include swapping rows, multiplying a row by a nonzero constant, and adding/subtracting multiples of rows.

• The solution can be unique, infinite, or nonexistent depending on the system.

Additional info: The file contains a set of questions guiding the student through the process of solving a system of equations using Gaussian elimination, identifying the coefficient matrix, and interpreting the solution.