The inverse matrix: undo what A does

Undo for numbers first

Multiplying by 3 is undone by multiplying by 1/3. The pair cancels and leaves 1, the number that changes nothing:

The number 0 has no undo: 0·x is 0 for every x, so from the answer 0 you can never tell what x was. Matrices work the same way, with one twist: many matrices besides the zero matrix (all entries 0) have no undo either.

The identity: the matrix that changes nothing

For matrices, the role of 1 is played by the identity matrix I from lesson 2: 1s on the diagonal and 0s everywhere else. I times any vector gives the same vector back:

So "A⁻¹ undoes A" means: A⁻¹ times A is I. Inverses only exist for square matrices (lesson 4 shows why), and for them the undo works both ways: AA⁻¹ = I too.

Example: undo “add the second entry to the first”

Take S = 1101. Multiply it by a vector pq, by rows as in lesson 1: S·pq = p + qq. S adds the second entry to the first and keeps the second entry. Start from 12:

To undo S, take the second entry away from the first one again: pq goes to p − qq. That is the matrix S⁻¹ = 1−101. Feed it the output 32:

That worked for one vector. To be sure it works for every vector, multiply the two matrices, as in lesson 2. The product should be I:

The lab draws each vector as an arrow from the origin, as in lesson 1. Multiplying by S gives a new arrow. Follow v = 12: S sends it to 32. Your S⁻¹ should bring it back.

Why the inverse solves equations

Start with Ax = b and multiply both sides on the left by A⁻¹. Since A⁻¹A = I and Ix = x, the left side is just x:

Try it with S. To solve Sx = 52, multiply 52 by S⁻¹:

If the inverse exists, this x is the one and only solution.

When there is no inverse

Some matrices can't be undone. Here is one. Its second column is just twice the first:

Feed it two different inputs:

Different inputs, same output. If all you see is 24, you can't tell which input made it. So nothing can undo A, and A has no inverse. The same happens with A·2−1 = 2 − 24 − 4 = 00: this input lands on 00, just like the input 00 does.

A matrix that has an inverse is called invertible. For a square matrix, each of these says the same thing: A is invertible.

• Every pivot is nonzero (lesson 2). For A above, row 2 − 2·row 1 gives 00, so the second pivot is 0.
• No column is a combination of the others. Otherwise the outputs miss part of the space (in the plane: they all land on one line, as in lesson 1's column space), so some outputs are shared. Lesson 4 calls columns like that, with no column a combination of the others, independent. For A, column 2 = 2·column 1.
• Ax = 0 only when x = 0. For A, x = 2−1 also gives 0.

So one failed test means all of them fail. The next slide adds a quick number test for 2 by 2 matrices.

A bigger example: a 3 × 3 inverse

Lesson 2's elimination matrix E₂₁ subtracts 2 times row 1 from row 2. Its undo adds 2 times row 1 back:

A matrix with zeros everywhere off the diagonal is called a diagonal matrix, often named D. It multiplies each entry of a vector by its own number, so its inverse divides by those numbers:

The idea is the same for a matrix with a thousand rows: A⁻¹ is the matrix whose product with A is I.

Next: How do we actually find A⁻¹? For a 2 by 2 matrix there is a short recipe, and one number tells at once whether the inverse exists.