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.

# Is option 3 also correct ? why so

**prateek111**#1

**Harsha_1997**#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