If A and B are square matrices of order 2 such that det(AB) = det(BA), then which one of the following is correct?
A and B need not be unit matrices.
Key property of determinants: For any two square matrices of the same order, the determinant of their product equals the product of their determinants, i.e. det(AB) = det(A) · det(B).
Apply the same property to the product BA: det(BA) = det(B) · det(A).
Since scalar multiplication is commutative, det(A) · det(B) = det(B) · det(A). Therefore the two expressions are equal for every pair of square matrices A and B:
det(AB) = det(A) · det(B) = det(B) · det(A) = det(BA).
Because the equality det(AB)=det(BA) holds for all square matrices of the same order, it places no restriction that would force A or B to be the identity matrix.
Conclusion: The correct choice is Option 3 — A and B need not be unit matrices.
Remark: This argument is valid for square matrices of any size, not only 2×2.
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