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Let A be a 4 x 4 matrix.

a) If the eigenvalues of A are 1,-2,3,-3, is it possible to determine det(A)? Why or why not?

b) What if the eigenvalues are -1,1,2?

c) What if the eigenvalues are -1,0,1?

I remember reading somewhere that the det(A) is equal to the product of all the eigenvalues of A but why is that so? If what I just said is true, wouldn't there be lack of information to calculate the determinant for parts b and c?

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To answer your question you need the equations $det(AB)=det(A)det(B)$ and $det(A^{-1})=det(A)^{-1}$.

In the first case you know that the matrix is diagonalizable (since there are 4 distinct eigenvalues). Hence $det(A)=det(SDS^{-1})=det(D)$, in this case the product of all eigenvalues.

As you noted correctly, in cases b) there is a lack of information. Even if the matrices are diagonal you can easily make up examples with different determinants.

EDIT: In case c) you have $0$ as an eigenvalue, this means that the matrix is non-invertible and the determinant is zero.

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    Sure, my fault. I editted.2012-10-20