Inverse of a product: undo the steps in reverse order

Socks and shoes with row steps

Take two row steps, like those in lesson 2. Each one is a matrix acting on the left, like lesson 2's elimination matrices. Step E: row 2 − 3·row 1. Step F: row 1 + 2·row 2. Do E first, then F, on the matrix 1235. Then undo them: first undo F (row 1 − 2·row 2), then undo E (row 2 + 3·row 1).

Now undo them in the wrong order, undo-E first:

The same rule with matrices

Matrix products act right to left: the step nearest the matrix acts first, so E then F is written FE. The right undo was E⁻¹F⁻¹. In general, in ABx, B is next to x, so B acts first. A went on last, so A⁻¹ must act first. In B⁻¹A⁻¹ the factor on the right, A⁻¹, acts first, which is what we need. The algebra check:

A meets A⁻¹ in the middle, and they cancel. Then B and B⁻¹ cancel. In the other order, A⁻¹B⁻¹AB, nothing meets its own inverse, and the result is usually not I.

A worked example: two simple steps

B = 1101 adds the second entry to the first (slide 1). A = 2001 doubles the first entry. Do B first, then A:

The wrong order, A⁻¹B⁻¹, fails:

Another example: R and S from slide 1

A bigger example: three 3 × 3 steps

Lesson 2 eliminated with three steps, E₃₂E₃₁E₂₁A = U. E₂₁ subtracts 2·row 1 from row 2, E₃₁ adds row 1 to row 3, and E₃₂ adds row 2 to row 3. To undo all three, reverse the order and undo each one:

That is lesson 2's L, the matrix in A = LU. Each undo flips the sign of its multiplier, and in this reversed order the multipliers drop into L without mixing (see lesson 2, slide 7). A computer chaining a thousand steps undoes them the same way: last step first.

Where this is used

A robot arm chains joint moves, one after another. To reverse them, undo the last joint first.

Next: The 2 by 2 recipe does not work for bigger matrices. Slide 4 finds any inverse with elimination, by working on A and I side by side.