Indeterminate determinants

I was marking the work of an otherwise quite able student today, and came across an offering that can be summarised thus:

We need to calculate \mathrm{det}(A). Therefore we need to find the eigenvalues of A. Therefore we need to solve the equation \mathrm{det}(A-\lambda I) = 0. Therefore…

To do justice to this student, s/he did realise at this point that he was heading in the general direction of the herring sandwich experiment, and stopped. It would have been nice if s/he could have gone back to the definition of the determinant and calculated it, but I suspect all this was going on shortly before the deadline in any case.

What this underlines for me, though, is the power of conditioning. I’d guess that on almost every other recent occasion when this student has had to calculate a determinant, it’s been in order to find eigenvalues. Thus, procedurally, a call to “calculate determinant” automatically triggers “find eigenvalues” by the power of association: the determinant has no independent existence as object or as process. Not an unusual syndrome, to be sure, but a nice clean example of it.

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