Let A be a skew-symmetric matrix of order 3. What is the value of det(4A^{4})-det(3A^{3})+det(2A^{2})-det(A)+det(-I) where I is the identity matrix of order 3?

Explanation

For a skew-symmetric matrix of odd order, det(A) = 0. Therefore, any expression containing det(A^n) = (det(A))^n is 0. Thus, det(4A^4)=0, det(3A^3)=0, det(2A^2)=0, det(A)=0. Also det(-I) = (-1)^3 det(I) = -1. Substituting these gives 0 - 0 + 0 - 0 - 1 = -1.

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