Gaussian Elimination: Row Reduction, Solutions, and Matrix Applications

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What is Gaussian elimination?

An algorithm that applies elementary row operations to a matrix to simplify a linear system, typically toward row echelon form.

What is an augmented matrix?

A matrix that places the coefficients of a linear system and its constants together, with the constants in the final column.

What are the three elementary row operations?

Swap two rows, multiply a row by a nonzero scalar, and add a multiple of one row to another.

Why do row operations preserve solutions?

Each operation replaces equations with equivalent equations and is reversible, so the same variable values satisfy the system.

What does $R_i \to cR_i$ mean?

Replace row $i$ by $c$ times that row, where $c$ must be nonzero.

Quiz questions

1. Which additional condition distinguishes reduced row echelon form from ordinary row echelon form?Answer: Each pivot is 1, with zeros elsewhere in its column.
2. What can be concluded about a consistent linear system with at least one free variable?Answer: It has infinitely many solutions.
3. How does swapping two rows affect the determinant of a square matrix?Answer: It multiplies the determinant by $-1$.
4. When a square matrix $A$ is invertible, what result follows from reducing $[A\mid I]$?Answer: The block matrix becomes $[I\mid A^{-1}]$.
5. In partial pivoting, which entry is chosen for the current pivot position?Answer: The available entry with the largest absolute value in that column.