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title: Invert, always invert
modified: 2026-07-19
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In the 19th century, mathematician [Carl Gustav Jacob Jacobi](https://www.google.com/search?q=Carl+Gustav+Jacob+Jacobi) famously said:

> Man muss immer umkehren.

which translates to _“You must always invert.”_

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In **linear algebra**, proofs often require working with the inverse of a matrix to arrive at the solution. However, determining the inverse itself usually starts with the end goal in mind: verifying properties like $A^{-1}A = I$ or solving $AX = B$, the steps to constructing or applying the inverse become more focused and straightforward.

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# Inversion

* Complex problems are sometimes better solved backwards
* Invert the problem
* Prevention over pursuit
* Focusing on what to avoid rather than what to achieve
* Reverse Engineering

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By identifying what we want to _avoid_, we often gain clearer insight into what we should actually _do_. The negative space defines the positive space.

# Examples

* **Investing**:
	* “How do I pick winning stocks?” ➞ “How do I avoid losing money?”
	* “How much money can I make” ➞ “How can I lose not as much money?”
* **Product Design**: Instead of “What features should I add?”, ask “What would make users hate this product?”
* **Health & Fitness**: Instead of “How do I get in shape?”, ask “What habits would guarantee I stay unhealthy?”
* **Decision-Making**: Before committing to a major choice, run a _[premortem](premortem.md)_: imagine it is a year later and the decision has failed catastrophically — what went wrong?

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| Instead of asking…                   | Ask…                                        |
| ------------------------------------ | ------------------------------------------- |
| How do I achieve X?                  | What would guarantee failure?               |
| How do I win?                        | How do I ensure I don’t lose?               |
| How do I get rich?                   | What would guarantee I go broke?            |
| How do I be productive?              | What would make me completely unproductive? |
| How do I build a great relationship? | What would ruin a relationship?             |
| How do I find happiness?             | What would make me miserable?               |
| How do I stay healthy?               | What would destroy my health?               |

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[No unforced errors](no-unforced-errors.md)