11.6 Solving Systems with Gaussian Elimination

Gaussian Elimination

Augmented Matrices from Equations

An augmented matrix is just a compact way to write a system of linear equations. Instead of writing out all the variables and equals signs, you organize the numbers into a grid.

Each row represents one equation. The entries in each row are the coefficients (the numbers multiplying each variable), written in the same variable order for every equation. The last column, separated by a vertical line, holds the constants from the right-hand side.

To build an augmented matrix from a system:

Say you have:

$2x + 3y - z = 5$

$x - y + 4z = 8$

$3x + 2y + z = 1$

1. List the coefficients of each variable in the same order ($x$, $y$, $z$) for every equation
2. Place the constants from the right-hand side in the last column
3. Separate the coefficient columns from the constant column with a vertical line

That system becomes:

$\left[\begin{array}{ccc|c} 2 & 3 & -1 & 5 \\ 1 & -1 & 4 & 8 \\ 3 & 2 & 1 & 1 \end{array}\right]$

To go the other direction (matrix back to equations), read each row as an equation: the entries multiply the variables in order, and the last column gives the constant after the equals sign.

Row Operations for Echelon Form

The goal is to reshape the matrix into row echelon form, which makes the system easy to solve. A matrix is in row echelon form when:

• The first non-zero entry in each row (called the pivot) is 1
• Each pivot sits to the right of the pivot in the row above it
• Every entry below a pivot is zero

You're allowed exactly three types of elementary row operations to get there:

1. Swap two rows ($R_i \leftrightarrow R_j$)
2. Multiply a row by a non-zero constant ($kR_i \rightarrow R_i$)
3. Add a multiple of one row to another ($R_i + kR_j \rightarrow R_i$)

These operations change the appearance of the matrix but never change the solutions of the system.

Process for reaching row echelon form:

1. Focus on the first column. Use row operations to make the top-left entry equal to 1 (this is your first pivot). Swapping rows can help if the top-left entry is 0.
2. Use the third row operation to make every entry below that pivot equal to 0.
3. Move to the next column and the next row down. Repeat: create a pivot of 1, then eliminate everything below it.
4. Continue until the matrix is in row echelon form.

Interpreting Row Echelon Form

Once your augmented matrix is in row echelon form, you can classify the system:

• Unique solution: There's a pivot in every variable column (every column except the last). You can solve for each variable exactly, typically using back-substitution (start from the bottom row and work upward).
• Infinitely many solutions: At least one variable column has no pivot. That variable is called a free variable, and you express the other variables in terms of it. For example, if $z$ is free, your answer might look like $x = 3 - 2z$, $y = 1 + z$, $z = z$.
• No solution: You see a row where all the coefficient entries are 0 but the constant is non-zero, like $[0 \quad 0 \quad 0 \mid 5]$. That row says $0 = 5$, which is impossible. The system is inconsistent.

Gaussian Elimination for Systems

Gaussian elimination is the full procedure that ties everything above together.

Steps:

1. Write the system of equations as an augmented matrix
2. Use row operations to transform the matrix into row echelon form
3. Interpret the result to classify the system (unique, infinite, or no solution)
4. If a unique solution exists, use back-substitution: solve the bottom equation first, then substitute upward

For a two-variable system, your row echelon form will have two rows. The bottom row gives you one variable directly, and you plug it into the top row to find the other.

For a three-variable system, the bottom row gives one variable, the middle row (after substitution) gives the second, and the top row gives the third. If there's a free variable, express the dependent variables in terms of it.

Linear Systems and Matrices

A linear system is a set of equations where every variable appears only to the first power (no $x^2$, no $xy$, no $\sqrt{x}$). A matrix is a rectangular array of numbers used to represent such a system compactly. The variables in the system are the unknowns you're solving for, and the matrix captures all the numerical information you need to find them.