Consider the following statements: S1: The sum of two singular n x n matrices may be non-singular. S2: The sum of two non-singular n x n matrices may be singular. Which one of the following statements is correct? A. S1 and S2 both are true B. S1 is true, S2 is false C. S1 is false, S2 is true D. S1 and S2 both are false
GATE 2001 · Engineering Mathematics · Matrix · medium
Answer: Both S1 and S2 are true. Answer: A.
- Test S1: sum of two singular matrices may be non-singular: Let A = [[1,0],[0,0]] and B = [[0,0],[0,1]]. Both are singular (det = 0). A + B = [[1,0],[0,1]] = I, which has det = 1 != 0. So S1 is TRUE.
- Test S2: sum of two non-singular matrices may be singular: Let A = [[1,0],[0,1]] (I) and B = [[-1,0],[0,-1]] (-I). Both non-singular with det = 1 and det = 1. A + B = [[0,0],[0,0]], which has det = 0 (singular). So S2 is TRUE.