For any invertible matrix A with det(A) ≠ 0, which statement is true about det(A^{-1})?

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Multiple Choice

For any invertible matrix A with det(A) ≠ 0, which statement is true about det(A^{-1})?

The important fact is how determinants behave with inverses. Since A is invertible, multiplying A by its inverse gives the identity: A A^{-1} = I. Taking determinants of both sides and using det(AB) = det(A) det(B) gives det(A) det(A^{-1}) = det(I) = 1. Because det(A) ≠ 0, you can solve for det(A^{-1}) by dividing: det(A^{-1}) = 1 / det(A). So the determinant of the inverse is the reciprocal of the determinant of A. For example, if det(A) = 6, then det(A^{-1}) = 1/6. The other possibilities don’t fit this fundamental relation.

The important fact is how determinants behave with inverses. Since A is invertible, multiplying A by its inverse gives the identity: A A^{-1} = I. Taking determinants of both sides and using det(AB) = det(A) det(B) gives det(A) det(A^{-1}) = det(I) = 1. Because det(A) ≠ 0, you can solve for det(A^{-1}) by dividing: det(A^{-1}) = 1 / det(A). So the determinant of the inverse is the reciprocal of the determinant of A. For example, if det(A) = 6, then det(A^{-1}) = 1/6. The other possibilities don’t fit this fundamental relation.

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