Strang §2.5 · Gauss-Jordan elimination
Eliminate on [A | I] and the inverse appears.
Write A and the identity side by side. Use row operations to turn A into I. The same operations turn I into A⁻¹.
[ A | I ] → [ I | A⁻¹ ]
Think: carbon paper
Put a sheet of carbon paper under your work. Every row operation you do on A gets copied onto the identity beside it.
When A has been turned into I, the copy holds the record of everything you did. Undoing A is exactly what that record does, so the copy is A⁻¹.
The steps
- Forward: clear below the pivots, as in lesson 2.
- Backward: clear above the pivots too, from the bottom up.
- Scale: divide each row by its pivot so the diagonal becomes 1s.
The result is the reduced row echelon form. The lab's rref does all three steps.
Worked example
K = 2−10−12−10−12 shows up for a chain of springs and masses.
K⁻¹ = ¼ · 321242123
Check one column: K times (3, 2, 1) = (6 − 2, −3 + 4 − 1, −2 + 2) = (4, 0, 0), which is 4 times the first column of I ✓
Don't use it to solve Ax = b
Computing A⁻¹ costs about three times the work of elimination, and it adds rounding error. To solve Ax = b, use elimination (A = LU). As Strang stresses: the inverse is great for thinking, but don't compute A⁻¹ to solve Ax = b.