The transpose of a matrix is the operation of switching rows and columns. Let r be a real number and A and B be matrices. Then, which of the following property does not hold ?

I. (A^T)^T = A
II. (A + B)^T = B^T + A^T

III. (AB)^T = B^TA^T

IV. (rA)^T = rA^T
(A) Only II
(B) Both II and IV
(C) None
(D) All


Answer: (C)

Explanation: I. (A^T)^T = A
II. (A + B)^T = A^T + B^T = B^T + A^T
III. (AB)^T = B^TA^T
IV. (rA)^T = rA^T

All given properties hold.

Option (C) is correct.

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