Strang §2.5 · The 2 by 2 inverse
Swap, flip signs, divide by the determinant.
For a 2 by 2 matrix there is a short recipe. Swap the two diagonal entries, change the signs of the other two, and divide everything by ad − bc.
abcd⁻¹ = 1/(ad − bc) · d−b−ca
Think: why divide by the determinant?
Lesson 1 showed that A multiplies every area by |det A|, where det A = ad − bc (its sign records a flip). To undo A, the inverse has to scale areas back by 1/|det A|, and its determinant is 1/det A.
When det A = 0, A squashes the plane flat. You would need to divide by zero to get back. No matrix can unflatten a line into a plane, so there is no inverse.
Worked example
A = 2153 det A = 2·3 − 1·5 = 1
A⁻¹ = 1/1 · 3−1−52
Check: AA⁻¹ = 6 − 5−2 + 215 − 15−5 + 6 = 1001 ✓
Watch out
- Swap the diagonal (a and d). Do not swap b and c.
- Flip the signs of b and c only.
- Always check by multiplying. AA⁻¹ should give I, and so should A⁻¹A.
Beyond 2 by 2
There is a formula for bigger matrices too, but it is slow. In practice the inverse comes from elimination. Slide 4 shows how.