An augmented matrix is a compact way to write a system of linear equations by placing the coefficients and constants into one rectangular array. It lets you focus on the numbers while preserving the structure of the equations. Row operations are the legal moves that transform the system without changing its solution set.
This method is important because it gives a reliable path for solving many equations at once, especially when substitution becomes messy.
The goal is often to turn the augmented matrix into reduced row echelon form, where solutions can be read directly from the final columns. Each row operation represents adding equations, scaling equations, or reordering equations in a way that keeps the system equivalent. A vertical bar is used to separate coefficient columns from the constant column.
When the left side becomes an identity matrix, the right side gives the values of the variables.
Key Facts
- An augmented matrix for ax + by = c and dx + ey = f is [a b | c; d e | f].
- Row swap: Ri <-> Rj exchanges two rows without changing the solution set.
- Row scaling: Ri -> kRi, where k != 0, multiplies one row by a nonzero constant.
- Row replacement: Ri -> Ri + kRj adds a multiple of one row to another row.
- Reduced row echelon form has leading 1s, zeros above and below each leading 1, and leading 1s move right as rows go down.
- If the final form is [I | b], then the solution vector is x = b.
Vocabulary
- Augmented matrix
- An augmented matrix is a matrix that contains the coefficients of a linear system together with its constant column.
- Elementary row operation
- An elementary row operation is one of the three allowed row changes that preserves the solution set of a linear system.
- Pivot
- A pivot is a leading nonzero entry used to eliminate other entries in its column.
- Row echelon form
- Row echelon form is a matrix form where nonzero rows are above zero rows and each leading entry is to the right of the one above it.
- Reduced row echelon form
- Reduced row echelon form is row echelon form with each pivot equal to 1 and zeros everywhere else in each pivot column.
Common Mistakes to Avoid
- Multiplying a row by 0 is wrong because it destroys information and is not an allowed elementary row operation.
- Changing only one side of the augmented bar is wrong because each row represents a full equation, so coefficients and constants must be transformed together.
- Forgetting to clear entries above a pivot is wrong because row echelon form is not necessarily reduced row echelon form.
- Reading a row like [0 0 | 5] as a variable value is wrong because it means 0 = 5, so the system has no solution.
Practice Questions
- 1 Solve the system using an augmented matrix: x + y = 7 and 2x - y = 5.
- 2 Use row operations to reduce the augmented matrix [1 2 | 8; 3 -1 | 1] and find x and y.
- 3 A reduced augmented matrix contains the row [0 0 0 | 4]. Explain what this row means for the solution set of the original system.