The Universe of Disco

Show how the student could have solved it

A few days ago I offered these maxims about pedagogy:

1. It's not enough to show the student the answer; you should try to show them how to find the answer.

2. It's not enough to show the student how you can find the answer; you should try to show them how they could have found the answer.

A nice illustration popped up on Math SE this morning. OP asks:

If all eigenvalues of a matrix are 0 or 1, does that imply the matrix is idempotent?

Shortly afterward a comment from PrincessEev said, opaquely:

The matrices $$\left[\begin{matrix} 0&x\\ 0&0 \end{matrix}\right]$$ are obvious counterexamples for !!x\ne 0!!.

Uh, they are? It wasn't obvious to me. I mean, I think I see why the eigenvalues must be zero, without doing the calculation. But where did this example come from?

But then later they redeemed themselves by adding another comment:

it was just my first instinct to try a few examples with what felt like a bold claim: matrices with enough well-placed zeroes tend to vanish when raising them to powers

I understand now! Yeah, I could have thought of that, but didn't. So the second comment actually taught me something, not what the answer is, which not very useful, because who cares?, but how to find the answer, which contains knowledge that might be generally useful.

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