While saying hello to the professors and those interested in mathematical sciences, I wanted to know from the perspective of the history of mathematics, what factor or process of solving what problem caused the definition of determinants in matrices? Were determinants created only to understand the independence or dependence of vectors, or by understanding the determinants of a matrix, Can you understand other questions related to the matrix in question? If the answer is yes, what other things can be guessed or obtained from the determinant value of a matrix? Thanks
while I am not 100% sure (ask any AI with repeated prompts!), i think it is solution of system of linear equations. For 2 equations, go through TL>BR and TR>BL and minus, that's the determinant. same is for 3 equations. for 4 or more, determinants are not as clearly intuitive as 3*3 matrix, but form of their solution would have triggered the cause of standardization of determinants. After all,-N by N matrixs' determinants are (carefully, signed) sum of product of every concievable terms not belonging to same row ar column- this is not what easily clicks anyone's mind. It is possible that cayley-hamilton theorem has generatlized determinant further to exterior products of matrices. It is also possible that product of a matrix and its adjoint matrix, as it arose to solve such systems of linear equations concretized the idea of a unique determinant for every square matrix.
The "full" history might be seen, e.g., next
https://mathshistory.st-andrews.ac.uk/HistTopics/Matrices_and_determinants/
... beginning with the Babylonians and Chinese 300-200 BC.
And, specifically about the term "determinant" it is said that ...
The idea of a determinant appeared in Japan before it appeared in Europe. In 1683 Seki wrote Method of solving the dissimulated problems which contains ...
Good luck with the reading
English mathematician, James Sylvester (1814–1897) invented matrix maths. In his 1850 dated paper, Sylvester stated, “For this purpose we must commence, not with a square, but with an oblong arrangement of terms consisting, suppose, of m lines and n columns. This will not in itself represent a determinant, but is, as it were, a Matrix out of which we may form various systems of determinants by fixing upon a number p, and selecting at will p lines and p columns, the squares corresponding of pth order.” Sylvester used the term matrix. Sylvester’s collaborator, English mathematician Arthur Cayley (1821–1895), used a form of matrices in his work. As it is known both matrices and determinants are used to solve linear equations, vector maths, statistics etc.
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