Augmented Matrices

An augmented matrix is simply a system of equations written as a grid of its numbers, with the letters left out. A matrix is any rectangular grid of numbers; "augmented" means "added to," because the right-hand sides are tacked on as an extra column after the bar. All we're really doing is elimination, same as the last two lessons, but without writing $x$, $y$ and $z$ over and over. The bracket up top looks new, but read it piece by piece: each row (one equation), each column (one variable's coefficients), and the bar (where the equals sign used to be). Nothing else has changed. It's honestly less writing, not more.

Row operations. When you did elimination by hand, you only ever did three things to equations. Here they are again, now as things you do to rows:

1. Swap two rows.
2. Multiply a row by a nonzero number.
3. Add a multiple of one row to another: $R_2 \to R_2 - 3R_1$.

None of these change the solutions. Swapping just reorders the list. Multiplying both sides of an equation by a nonzero number keeps it true. And adding a multiple of one true equation to another gives another true equation, which is exactly why elimination worked.

The goal is row-echelon form: 1s down the diagonal and 0s below. The bottom row then has just one variable, so you solve it and back-substitute upward. If you keep going and get 0s above the diagonal too (reduced row-echelon form), the solution is sitting right there in the last column.

Reading special rows. Remember the two strange endings from elimination? They show up here too.

• $[\,0\ 0 \mid c\,]$ with $c \ne 0$ means $0 = c$: no solution.
• A row of all zeros means one equation was redundant: often infinitely many solutions.

This is exactly how computers solve systems, and it's where linear algebra begins.

Circuit analysis, structural engineering, computer graphics and machine learning all reduce to large linear systems solved by row reduction.