Just wondering about the structure of a determinant

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Just wondering about the "structure" of a determinant...

How much can a determinant tell you about the entries of a matrix? How much more if you know the size of the aforementioned matrix? How much more if you know that the matrix is symmetric?(perhaps a silly question). How much more if you know the cofactors across the ith row or jth column?
 
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Very little. An n by n matrix contains n^2 entries and taking the determinant reduces all that information to one number. You will have lost an immense amount of information that you cannot get back just knowing the determinant.

The one thing you can be sure about, just by knowing the determinant of a matrix, is whether it is invertible or not.
 


The determinant is the factor by which volumes are multiplied under the linear transformation that the matrix represents. That is all it tells you.
 


In a complex vector space the determinant is just the product of the eigenvalues. Sometimes that is helpful, often times not. Likewise, the fact that the trace of a matrix is the sum of the eigenvalues is also sometimes of use.
 

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