Is option 3 also correct ? why so


#1

Two matrices A and B are called similar if there exists another matrix S such that S−1AS = B. Consider the statements:
(I) If A and B are similar then they have identical rank.
(I) If A and B are similar then they have identical trace.
(III) A = 1 0 B = 1 0
0 0 1 0
Which of the following is TRUE.


#2

Two n-by-n matrices A and B are called similar if

B = S^−1AS

for some invertible n-by-n matrix S.

Similar matrices share many properties:

Rank
Determinant
Trace
Eigenvalues (though the eigenvectors will in general be different)
Characteristic polynomial
Minimal polynomial (among the other similarity invariants in the Smith normal form)
Elementary divisors