Suppose A and B are similar real matrices, that is, there exists an invertible matrix S such that $$A = S B S^{-1}.$$​
Which of the following need not be true?

Correct option is D

​$$\text{Given } A \sim B, \text{ we know that:}\\[10pt]A^T \sim A \quad \text{and} \quad B^T \sim B.\\[10pt]\text{Thus:}\\[10pt]A^T \sim B^T \quad \text{(Statement in Option A is true).}\\[10pt]$$

​​Since similar matrices have the same eigenvalues, characteristic polynomial, and minimal polynomial:

Statements in Options B and C are also true.

However, the similarity relation does not say anything about the range.

Therange of A may not be equal to the range of B.

Thus, the statement in OptionD need not be true.