Augmented Matrices

When you solve a system by elimination, the letters just come along for the ride. Every step changes the numbers, but , and sit in the same columns the whole time. An augmented matrix drops them and keeps only the numbers, arranged in a grid. That makes the work shorter and harder to get wrong, and it’s exactly how graphing calculators and computers solve systems. This lesson shows how to write a system as an augmented matrix and how to solve it with Gaussian and Gauss-Jordan elimination.

What Is an Augmented Matrix?

A matrix is a rectangular grid of numbers. The augmented matrix of a linear system has:

• one row for each equation,
• one column for each variable, in the same order in every row,
• and one extra column for the constants, set off by a vertical bar that stands in for the equals signs.

The same system written both ways:

A variable with no visible number in front has coefficient , and a subtraction sign means a negative entry: gives and .

Coefficient Matrix vs Augmented Matrix

The coefficient matrix is just the part to the left of the bar, the coefficients of the variables. The augmented matrix “augments” it with the column of constants:

Which Matrix Represents the System of Equations?

Writing the matrix correctly is mostly about lining things up. Two things trip people up: missing variables and equations that aren’t in standard form.

Write the augmented matrix for:

Step 1 — put every equation in standard form, with variables on the left in the order :

Step 2 — copy the coefficients, using for every missing variable:

The zeros matter. Leaving a gap instead of a shifts the numbers into the wrong columns, and the matrix then represents a different system.

Which System Represents the Matrix?

Going the other way is just as quick: attach the variables to the columns and put the equals sign back where the bar is. The matrix

represents

A in a column means that variable doesn’t appear in that equation.

The Three Row Operations

To solve a system with its matrix, you transform the matrix step by step. Only three moves are allowed, and each one matches something you could do to the equations themselves without changing the solution:

The goal is to create zeros below (and, for Gauss-Jordan, above) the main diagonal, one column at a time.

Row-Echelon Form and Reduced Row-Echelon Form

Gaussian Elimination vs Gauss-Jordan Elimination

Both methods use the same row operations. Gaussian elimination stops at row-echelon form and finishes with back-substitution, which is the triangular-form technique from systems with three variables. Gauss-Jordan elimination keeps going until the left side is the identity matrix, so each row reads “variable = number”. Gaussian takes fewer row operations; Gauss-Jordan needs no back-substitution. On paper, use whichever you find tidier.

Example 1: Gauss-Jordan Elimination on a 2 × 2 System

Solve and .

Step 1 — write the augmented matrix:

Step 2 — make a zero below the first leading 1 with :

Step 3 — make the second leading entry 1 with :

Step 4 — clear the entry above it with :

Step 5 — read the answer: the rows say and . Check: ✓ and ✓.

Example 2: Gaussian Elimination on a 3 × 3 System

Solve , and .

Step 1 — the augmented matrix:

Step 2 — clear the first column with and :

Step 3 — swap so the second row starts with a 1:

Step 4 — clear below it with :

Step 5 — back-substitute from the bottom row up:

Answer: , the same solution the elimination method gives in the three-variable lesson, with a lot less writing.

No Solution and Infinitely Many Solutions in Matrix Form

The final matrix also tells you when there isn’t exactly one solution.

A row with means no solution. For and , the operation gives

and the second row says . The system is inconsistent.

A row of all zeros means infinitely many solutions (when there are as many equations as unknowns). For and , gives

Only one real equation is left, , so is a free variable. Let . The general solution is

Each value of gives one of the infinitely many solutions: gives , gives , and so on. An augmented matrix whose final form has no row of the form with belongs to a consistent system.

Common Mistakes to Avoid

• Skipping zeros for missing variables. is the row , not .
• Mixing up the column order. Every row must list the variables in the same order, usually .
• Applying an operation to only part of a row. changes every entry of , including the constant after the bar.
• Multiplying a row by zero. That deletes an equation. Scaling must use a non-zero number.
• Misreading the final matrix. In row-echelon form you still need back-substitution; only reduced row-echelon form shows the answer directly.

Practice Problems

Work each one before opening the answer.

Problem 1. Write the augmented matrix for and .

Problem 2. Which system of equations does this augmented matrix represent?

Problem 3. Write the augmented matrix for , and .

Problem 4. Solve and using an augmented matrix.

Problem 5. A system’s augmented matrix reduces to rows and . What is the solution?

Problem 6. A system’s augmented matrix reduces to rows and . How many solutions does it have?

Matrices work for any number of linear equations, but every equation so far has been a straight line or a flat plane. When one of the equations is a curve, such as a parabola or a circle, the system is nonlinear, and substitution becomes the main tool again.