Row Echelon Form Explained: Gaussian Elimination

This video is a step-by-step guide to understanding Row Echelon Form (REF) and Reduced Row Echelon Form (RREF) in linear algebra. We start by defining REF: zero rows at the bottom, pivots in each row, and pivots shifting right as you go down. Then we explain RREF: pivots equal to 1, and each pivot is the only nonzero element in its column. Through visual examples, we check if matrices satisfy the REF or RREF conditions and explain pivot columns and non-pivot columns. We move on to solving systems of linear equations using RREF, finding both particular solutions and general solutions. You’ll learn how basic variables are linked to pivot columns, and free variables are linked to non-pivot columns, which lead to infinite solutions. Finally, we show Gaussian elimination (Gauss-Jordan method) to convert any matrix into RREF, using row operations (row swaps, scaling, adding multiples of rows). By the end, you’ll know how to identify unique solutions, no solutions, or infinite solutions from the RREF of a matrix. 00:00 REF vs. RREF 02:20 How to solve systems of linear equations using RREF (Particular and General solutions) 06:58 Gaussian Elimination (Gauss-Jordan Method): How to find the RREF of any matrix to find the general solutions