Consider the following statements: S1: The sum of two singular n × n matrices…
2001
Consider the following statements:
S1: The sum of two singular n × n matrices may be non-singular.
S2: The sum of two n × n non-singular matrices may be singular.
Which of the following statements is correct?
- A.
S1 and S2 are both true
- B.
S1 is true, S2 is false
- C.
S1 is false, S2 is true
- D.
S1 and S2 are both false
Attempted by 10 students.
Show answer & explanation
Correct answer: A
Both statements are true because each asks whether the situation is possible.
For S1, take two 2 × 2 singular matrices:
A = [[1, 0], [0, 0]] and B = [[0, 0], [0, 1]].
Both A and B are singular, but A + B = [[1, 0], [0, 1]], which is non-singular.
For S2, take A = I and B = -I. Both I and -I are non-singular, but A + B = 0, the zero matrix, which is singular.
Therefore, both S1 and S2 are true. Hence, option A is correct.