Finding Row Echelon Form

Linear Algebra (Matrices) · Axiom Academy

EXAMPLE Finding Row Echelon Form Reduce a augmented matrix to row echelon form, choosing the correct row operation at each pivot. Use Gaussian elimination to put the following augmented matrix into row echelon form — leading 1 's descending in a staircase, with all zeros below each pivot. Nice work — you reduced the matrix to row echelon form one pivot at a time. Here is what to carry forward: Work left to right: place a pivot in the leftmost available column, then clear every entry below it before moving on. Choose the multiplier: to kill entry a_ ij with pivot a_ kj , use m = a_ ij /a_ kj and apply . Swaps position pivots: a row swap (here R1 ↔ R2) is a legal operation that can hand you a clean leading 1 to start. Leading 1 's: scale a pivot row (R3 → −R3, then R4 ÷ (−21)) so each pivot is exactly 1 — that is what makes it row echelon form. The staircase: the finished matrix has zeros below every pivot and each leading 1 strictly to the right of the one above it. From row echelon form you can back-substitute to solve the system, or press on with Gauss–Jordan elimination to reach reduced row echelon form. The same routine scales to a matrix of any size.

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