The absolute value (modulus) might be the most origin of determinant from a cultural and rather trivial point of view, indeed there is a deep connect to the very roots of valuation (and norm). In that perspective the origin of determinant can be sealed and determined, somehow trivially, by history of human kind when value and norms found meaning in even primitive measurements.
BC $300-200$ there is evidence from Babylonians and Chinese that they applied equation-solving via semi-matrices but it is unclear whether an interpretation of determinants can be justified, and rather not explicit enough.
This is however rather too scrutinized history.
But more fixed on the term determinant in math discipline, indeed the first appearance of a determinant in Europe was in 1683 in a note by Leibniz to de l'Hôpital, explaining that the system of equations
$$10 + 11x + 12y = 0$$
$$20 + 21x + 22y = 0$$
$$30 + 31x + 32y = 0$$
has a solution because
$$10\cdot21\cdot32 + 11\cdot22\cdot30 + 12\cdot20\cdot31 = 10\cdot22\cdot31 + 11\cdot20\cdot32 + 12\cdot21\cdot30$$
this is exactly the condition that the coefficient matrix has determinant 0. Leibnitz does not use nummerical coefficients but two characters, the first marking in which equation it occurs, the second marking which letter it belongs to.
The intuition and reasoning of application of determinants was originally tied with equation solving.
By the way it was Laplace who tailored the term and used the word resultant for what we today call determinant and the term determinant was first introduced by Gauss in Disquisitiones arithmeticae (1801) while discussing quadratic forms (see also here>>>).