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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
Which of the following statements is correct?
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
Which of the following statements is correct?
- a)S1 and S2 are both true
- b)S1 is true, S2 is false
- c)S1 is flase, S2 is true
- d)S1 and S2 are both false
Correct answer is option 'A'. Can you explain this answer?
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Verified Answer
Consider the following statements:S1: The sum of two singular n x n ma...
S1 is true
Consider two singular matrices
Sum of A and B is given as
However (A + B) is a non-singuiar matrix
So, S1 is true.
Now, consider two non-singuiar matrices
However (C + D) is a singular matrix. So, S2 is also true.
Therefore, both S1 and S2 are true.
Consider two singular matrices
Sum of A and B is given as
However (A + B) is a non-singuiar matrix
So, S1 is true.
Now, consider two non-singuiar matrices
However (C + D) is a singular matrix. So, S2 is also true.
Therefore, both S1 and S2 are true.
Most Upvoted Answer
Consider the following statements:S1: The sum of two singular n x n ma...
Statement Analysis:
We are given two statements regarding the sum of two matrices:
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.
S1 Analysis:
To analyze the statement S1, let's consider two singular n x n matrices A and B. Since both A and B are singular, it means that the determinant of each matrix is zero. Let's denote the determinants as det(A) = 0 and det(B) = 0.
Now, let's consider the sum of these two matrices, C = A + B. To determine if C is non-singular, we need to check if the determinant of C is non-zero. So, we need to find det(C).
Using the property of determinants, we know that det(A + B) = det(A) + det(B) when A and B are square matrices of the same order. Substituting the determinants of A and B, we have det(C) = 0 + 0 = 0.
Since det(C) = 0, it means that the sum of two singular n x n matrices is indeed singular. Therefore, statement S1 is false.
S2 Analysis:
To analyze the statement S2, let's consider two non-singular n x n matrices A and B. Since both A and B are non-singular, it means that the determinant of each matrix is non-zero. Let's denote the determinants as det(A) ≠ 0 and det(B) ≠ 0.
Now, let's consider the sum of these two matrices, C = A + B. To determine if C is singular, we need to check if the determinant of C is zero. So, we need to find det(C).
Using the property of determinants, we know that det(A + B) = det(A) + det(B) when A and B are square matrices of the same order. Substituting the determinants of A and B, we have det(C) ≠ 0 + 0 ≠ 0.
Since det(C) ≠ 0, it means that the sum of two non-singular n x n matrices is indeed non-singular. Therefore, statement S2 is true.
Conclusion:
From our analysis, we can conclude that statement S1 is false and statement S2 is true. Therefore, the correct answer is option 'A' - S1 and S2 are both true.
We are given two statements regarding the sum of two matrices:
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.
S1 Analysis:
To analyze the statement S1, let's consider two singular n x n matrices A and B. Since both A and B are singular, it means that the determinant of each matrix is zero. Let's denote the determinants as det(A) = 0 and det(B) = 0.
Now, let's consider the sum of these two matrices, C = A + B. To determine if C is non-singular, we need to check if the determinant of C is non-zero. So, we need to find det(C).
Using the property of determinants, we know that det(A + B) = det(A) + det(B) when A and B are square matrices of the same order. Substituting the determinants of A and B, we have det(C) = 0 + 0 = 0.
Since det(C) = 0, it means that the sum of two singular n x n matrices is indeed singular. Therefore, statement S1 is false.
S2 Analysis:
To analyze the statement S2, let's consider two non-singular n x n matrices A and B. Since both A and B are non-singular, it means that the determinant of each matrix is non-zero. Let's denote the determinants as det(A) ≠ 0 and det(B) ≠ 0.
Now, let's consider the sum of these two matrices, C = A + B. To determine if C is singular, we need to check if the determinant of C is zero. So, we need to find det(C).
Using the property of determinants, we know that det(A + B) = det(A) + det(B) when A and B are square matrices of the same order. Substituting the determinants of A and B, we have det(C) ≠ 0 + 0 ≠ 0.
Since det(C) ≠ 0, it means that the sum of two non-singular n x n matrices is indeed non-singular. Therefore, statement S2 is true.
Conclusion:
From our analysis, we can conclude that statement S1 is false and statement S2 is true. Therefore, the correct answer is option 'A' - S1 and S2 are both true.
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Consider the following statements:S1: The sum of two singular n x n matrices may be non-singularS2: The sum of two n x n non-singular matrices may be singularWhich of the following statements is correct?a)S1 and S2 are both trueb)S1 is true, S2 is falsec)S1 is flase, S2 is trued)S1 and S2 are both falseCorrect answer is option 'A'. Can you explain this answer?
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