The Forward Pass

Linear Algebra (Matrices) · Axiom Academy

Drive a pivot down the diagonal and clear the column beneath it — turning any matrix into a clean echelon staircase. The forward pass rewrites a matrix into row-echelon form . Watch the work matrix on the left: as zeros snap into place below the main diagonal , the leading entries line up into a descending staircase on the right. Everything under each step is zero — that is the whole target. Pivots sit on the staircase; every entry below them is 0 2. The Pivot Marches Down the Diagonal A pivot is the leading nonzero entry we anchor each elimination to. The forward pass takes them one at a time, down the diagonal — and for each pivot it only ever touches the entries directly below it in that column. Watch the pivot box step from a_ 11 to a_ 22 to a_ 33 , lighting up the column it is responsible for clearing. Owns column 1. It will clear the two entries beneath it. Owns column 2 (the part below row 2). Clears the one entry under it. Bottom of the staircase. Nothing lies below it, so the pass ends. Finish clearing a pivot's column before moving to the next pivot down. Every zero comes from a single row-replacement . To kill the entry below a pivot, find the multiplier — target divided by pivot — then subtract that many copies of the pivot row. Below, the 4 in row 2 is driven to 0 as is peeled away from R_2 , entry by entry. m = a_ 21 /a_ 11 = 4/2 = 2 . It is exactly what makes the target cancel. , applied across the whole row.

This is the written version of the interactive lesson above. See the full Linear Algebra (Matrices) course.