GATE | GATE-CS-2001 | Question 2

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


Answer: (A)

Explanation: Singular Matrix: A square matrix is singular if and only if its determinant value is 0.

S1 is True: The sum of two singular n × n matrices may be non-singular
It can be seen be taking following example. The following two matrices are singular, but their sum is non-singular.

M1 and M2 are singular
M1 =  1  1
      1  1
       
M2 =   1  -1
      -1   1

But M1+M2 is non-singular  
M1+M2 =  2  0
         0  2

S2 is True: The sum of two n × n non-singular matrices may be singular

M1 and M2 are non-singular
M1 =  1  0
      0  1
       
M2 =   -1  0
        0  -1

But M1+M2 is singular  
M1+M2 =  0  0
         0  0


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