653 views
1 votes
1 votes

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) 
Which of the following is TRUE. 

   
   
 

(A) Only I and III 

 

(B) Only II 

 

(C) Both I and II but not III 

 

(D) All of I, II and III

1 Answer

1 votes
1 votes

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

Related questions

2 votes
2 votes
1 answer
1
Neal Caffery asked Dec 11, 2016
5,312 views
If A3x3 is a matrix with |A| = 2. What is the determinant of Adj (Adj (Adj A))?
0 votes
0 votes
0 answers
2
Anuj Mishra asked Nov 11, 2018
148 views
0 votes
0 votes
1 answer
3