If A and B are square matrices of order 2 such that det(AB) = det(BA), then which one of the following is correct?

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  1. A must be a unit matrix.
  2. B must be a unit matrix.
  3. Both A and B must be unit matrices.
  4. A and B need not be unit matrices.

Answer (Detailed Solution Below)

Option 4 : A and B need not be unit matrices.
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Detailed Solution

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Explanation:

If A and B are square matrices of order 2 such that det(AB) = det(BA)

This property is true regardless of whether A and B are unit matrices or not. Therefore, the condition det(AB) = det(BA) does not imply that either A or B must be a unit matrix.

Option 4 is correct 

Additional Information Properties of determinants:

  • The determinant of a diagonal matrix is the product of the diagonal entries.
  • If A and B are both n×n matrices, then det(AB) = det(A) det(B).
  • For a n×n matrix A, det(kA) = kn det(A).
  • The determinant of a square matrix is the same as the determinant of its transpose.

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