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.
Important Questions from Types of Matrices
If \({\rm{A}} = \left[ {\begin{array}{*{20}{c}} 1&1&{ - 1}\\ 2&{ - 3}&4\\ 3&{ - 2}&3 \end{array}} \right]{\rm{\;and\;\;B}} = \left[ {\begin{array}{*{20}{c}} { - 1}&{ - 2}&{ - 1}\\ 6&{12}&6\\ 5&{10}&5 \end{array}} \right]\) then which of the following is/are correct?
1. A and B commute.
2. AB is a null matrix.
Select the correct answer using the code given below:Which one of the following matrices is an elementary matrix?
The matrix is \(\left[ {\begin{array}{c} 0&{ - 4 + i}\\ {4 + i}&0 \end{array}} \right]\)
How many distinct matrices exist with all four entries taken from (1, 2)?
If matrix \({\rm{A}} = \left[ {\begin{array}{*{20}{c}} {1 - {\rm{i}}}&{\rm{i}}\\ { - {\rm{i}}}&{1 - {\rm{i}}} \end{array}} \right]\) where \(\rm i = \sqrt {-1},\) then which one of the following is correct?
