Practice question · Multiple choice
Diagonalisation writes A as PDP⁻¹. What is the change-of-basis story that makes A¹⁰⁰ = PD¹⁰⁰P⁻¹ obvious?
Hints
- Write out (PDP⁻¹)(PDP⁻¹) and look at what sits in the middle.
- Ask what P⁻¹P equals.
Show the answer
C. P⁻¹ translates in, D scales, P translates back; the rest cancels
Why
Each adjacent P⁻¹P collapses to the identity, so a hundred sandwiches telescope into one, translate, scale a hundred times, translate back. Option 2 is false in general, and mistaking a telescoping cancellation for commutativity is the standard slip here.
Practise Matrix Diagonalisation
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