Augmented matrix
A matrix that stores a system's coefficients and its right-hand sides together, separated by a bar. Row operations on it do not change the solutions.
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Analogy
A shorthand notebook where you write only the numbers, not the letters.
Example
Write [[2, 1 | 5], [1, 3 | 5]].First column holds x-coefficients.Second holds y-coefficients.Last column holds the right-hand sides.
In practice
The tidy way to apply Gaussian elimination by hand or by computer.
Practice
Flashcards
- Name the three allowed row operations.Swap rows, scale a row by a non-zero number, add a multiple of one row to another.
Interview questions
- Solve 2x + y = 5, x + 3y = 5 using Gaussian elimination and check your answer.
- Augmented matrix [[2,1 | 5],[1,3 | 5]]
- R2 minus half of R1 gives [0, 2.5 | 2.5]
- y = 1
- Back-substitute: 2x + 1 = 5, x = 2
- Check both original equations
Questions people ask
Which row operations are allowed?
You may swap two rows, multiply a row by a non-zero number, or add a multiple of one row to another. Each is reversible and keeps the same solutions. Multiplying a row by zero is not allowed, because it destroys information.
Why do row operations keep the solutions the same?
Each operation matches a step you would use on the equations themselves, such as swapping their order or adding two equations. Any solution of the old system also solves the new one, and because you can undo each step, the reverse holds too.