Consider the following statements:S1: The sum of two singular n x n matrices may be non-singular.S2. The sum of two n x n non-singular matrices may be singular.Q. Which of the following statements is true?a)S1and S2 both are trueb)S1 is true and S2 is falsec)S1 is false, S2 is trued)S1 and S2 both are falseCorrect answer is option 'A'. Can you explain this answer?

Question

Answer

Answer:

Introduction:
The given question is about the sum of two matrices and whether the resulting matrix is singular or non-singular. We need to evaluate the truth value of two statements, S1 and S2, and determine which option is correct.

Statement S1:
S1 states that the sum of two singular n x n matrices may be non-singular.

Explanation of S1:
For a matrix to be singular, its determinant must be zero. Consider two singular matrices A and B, both of size n x n. Let's assume that both A and B have determinants equal to zero. Now, we take the sum of these matrices, denoted as C = A + B. The determinant of C is given by det(C) = det(A + B).

If det(A) = 0 and det(B) = 0, it does not necessarily mean that det(A + B) = 0. The determinant of a sum of matrices is not equal to the sum of determinants. Therefore, it is possible for the sum of two singular matrices to result in a non-singular matrix. Hence, S1 is true.

Statement S2:
S2 states that the sum of two n x n non-singular matrices may be singular.

Explanation of S2:
For a matrix to be non-singular, its determinant must be non-zero. Consider two non-singular matrices A and B, both of size n x n. Let's assume that both A and B have determinants not equal to zero. Now, we take the sum of these matrices, denoted as C = A + B. The determinant of C is given by det(C) = det(A + B).

If det(A) ≠ 0 and det(B) ≠ 0, it does not necessarily mean that det(A + B) ≠ 0. The determinant of a sum of matrices is not equal to the sum of determinants. Therefore, it is possible for the sum of two non-singular matrices to result in a singular matrix. Hence, S2 is true.

Conclusion:
Based on the explanations above, both S1 and S2 are true. Therefore, the correct option is (a) S1 and S2 both are true.

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