Practice question · Multiple choice
The inverse of a product reverses the order: (AB)⁻¹ = B⁻¹A⁻¹, not A⁻¹B⁻¹. Why does undoing a composition require going backwards through it?
Hints
- Multiply out (AB)(B⁻¹A⁻¹) and see what cancels.
- You put on socks then shoes. In what order do you take them off?
Show the answer
C. Because AB means do B then A, so undoing reverses the order.
Why
AB means apply B then A, so undoing it means undoing the most recent step first, socks then shoes going on, shoes then socks coming off. The algebra shows the mechanism: (AB)(B⁻¹A⁻¹) cancels from the inside out. A⁻¹B⁻¹ is perfectly well defined and simply wrong, corresponding to undoing B first.
Practise Matrix Inverses
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