How the determinant was discovered?

For 2×2 matrix, A=[a b, c d]
we want to find B such that AB = I
Let B = [e f, g h]
then ae+bg=1
af+bh=0
ce+dg=0
cf+dh=1
from (2) and (3), we have
f=-bh/a and e=-dg/c
Put e into the (1),
-adg/c + bg = 1
-adg + bcg = c
g(bc-ad) = c
g = c/(bc-ad)
And we find a repeating occurrence of ‘bc-ad’, named as the determinant.

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