Manipulating Equations

Linear Algebra (Matrices) · Axiom Academy

LESSON The Three Elementary Row Operations Swap, scale, replace — the only three moves you need to solve any linear system, and why each one leaves the answer untouched. The first operation simply exchanges two rows . On the equations this just rewrites them in a different order — and the order in which you list equations cannot change what satisfies them. Watch the two rows trade places while the solution sitting below stays fixed. Same two equations — the solution is unchanged 2. Scale a Row by a Nonzero Constant The second operation multiplies every entry of one row by the same constant c . On the equation that is multiplying both sides by c — which never changes the solution, provided . Watch row 1's entries grow as it is scaled by 2 , while row 2 and the solution hold still. Every entry of R_1 , including the augmented column, is multiplied by c . R_2 never moves — only the scaled row changes. Scaling by 0 wipes the row to , destroying a constraint — that loses solutions, so it is forbidden. Because , the move undoes cleanly with . x+2y=5 and its double 2x+4y=10 have exactly the same solution set — any pair that satisfies one satisfies the other. Here both leave fixed. 3. Replace a Row Using Another The third operation replaces a row by itself plus a multiple of another row . This is the workhorse of elimination: choosing c well drives an entry to zero. Watch -2R_2 flow into R_1 , turning into — while the solution stays put.

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