Key Idea: A matrix is a grid of numbers. For a 2×2 matrix, the determinant tells you whether it can be undone, and the inverse undoes it, for example to move a transformed point back.
| Item | Rule |
|---|---|
| Order | Rows × columns (rows first) |
| Determinant | det A = ad − bc |
| Inverse | A⁻¹ = (1 / det A) × ((d, −b), (−c, a)) |
| Check | A⁻¹A = AA⁻¹ = I |
| det A = 0 | No inverse; the matrix is singular |
Inverting A = ((4, 3), (2, 5)) by hand
- Determinant — 4(5) − 3(2) = 14, not zero, so an inverse exists.
- Swap and negate — Swap the diagonal, negate the off-diagonal: ((5, −3), (−2, 4)).
- Divide — A⁻¹ = (1/14)((5, −3), (−2, 4)).
Swap, negate, divide, but only when the determinant is not zero.
What is the order of a matrix with 2 rows and 3 columns? 2 × 3.
Find det of ((2, 6), (1, 3)). Can you invert it? 2(3) − 6(1) = 0, so no.
A scanner distorts a logo with A. What does A⁻¹ do? It moves every point back, so the original logo returns.
A GDC can give A⁻¹. Why still know the hand method? So you can show the steps when asked to find it by hand.