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I can understand the adjugate matrix and the motivation of that to find the inverse, but I can't see how this idea was invented by mathematicians. It's just brilliance or someone understand how the properties were conceived.

I trying to understand this property origins, but I can't explain how cofactor were created too:

$$\sum_{i = 1}^n a_{ij} C_{ik}= \begin{cases} \det(A) & \text{if } j = k \\ 0 & \text{if } j \neq k \end{cases}$$

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    $\begingroup$ Try to look at the inverse matrix by finding the indeterminate entries with the Cramer's rule (which is rather old mathematics), and the matrix of cofactors appears quite naturally. That said, this is not a research question, so I vote to close. $\endgroup$
    – Francesco Polizzi
    Commented Oct 7, 2015 at 8:55
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    $\begingroup$ Allen Knutson parked his rephrasing of the question together with an answer at plus.google.com/+AllenKnutson/posts/LgLxgxsXNAT. $\endgroup$
    – Christian Stump
    Commented Oct 7, 2015 at 12:19
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    $\begingroup$ I vote to reopen: meta.mathoverflow.net/a/2519/1 $\endgroup$
    – Anton Geraschenko
    Commented Oct 7, 2015 at 18:01
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    $\begingroup$ mathoverflow.net/a/89079/290 $\endgroup$
    – Qiaochu Yuan
    Commented Oct 7, 2015 at 19:42
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    $\begingroup$ (My answer linked above has no essential difference from Qiaochu's answer, linked directly above.) $\endgroup$
    – Allen Knutson
    Commented Oct 7, 2015 at 21:26

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