Linear Algebra · Lesson 7
Systems of Equations via Row Reduction
Gaussian elimination is the systematic algorithm for solving any system of linear equations. By applying row operations to an augmented matrix, we reduce it to a form where the solution is easy to read off.
Quick Check
1. Which row operation changes the solution set?
Swapping two rows
Multiplying a row by a nonzero scalar
Adding a multiple of one row to another
None of the above — all row ops preserve the solution
2. A row [0, 0 | 5] in an augmented matrix means:
x = 5/0 = undefined
No solution (contradiction: 0 = 5)
Infinite solutions
x = 0
3. A row [0, 0 | 0] in an augmented matrix means:
No solution
A free variable (infinite solutions possible)
x = 0, y = 0
Unique solution
4. In RREF, each pivot column has:
All 1s
A single 1 and all other entries 0
A 1 in the last row only
The largest value as the pivot