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Let A = [ajj] be an n x n matrix with real entries such that the sum of all the entries in each row is zero. Consider the following statements
(I) A is non-singular
(II) A is singular
(III) 0 is an eigenvalue of A
(I) A is non-singular
(II) A is singular
(III) 0 is an eigenvalue of A
Which of the following is correct?
- a)Only (I) is true
- b)(I) and (III) are true
- c)(II) and (III) are true
- d)Only (III) is true
Correct answer is option 'C'. Can you explain this answer?
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Most Upvoted Answer
Let A = [ajj] be an n x n matrix with real entries such that the sum o...
Solution:
To determine whether the given matrix A is non-singular or singular and whether 0 is an eigenvalue of A, we can use the properties of the matrix and its row sums.
Properties of A:
1. A is an n x n matrix with real entries.
2. The sum of all the entries in each row is zero.
Statement (I): A is non-singular
To determine whether A is non-singular, we need to check if the determinant of A is nonzero. If the determinant is nonzero, then A is non-singular. Otherwise, A is singular.
Statement (II): A is singular
To determine whether A is singular, we need to check if the determinant of A is zero. If the determinant is zero, then A is singular. Otherwise, A is non-singular.
Statement (III): 0 is an eigenvalue of A
To determine whether 0 is an eigenvalue of A, we need to check if the equation Av = 0 has a nontrivial solution. If the equation has a nontrivial solution, then 0 is an eigenvalue of A. Otherwise, 0 is not an eigenvalue of A.
Explanation:
Since the sum of all the entries in each row of A is zero, we can say that the rows of A are linearly dependent. This implies that the rank of A is less than n.
Since the rank of A is less than n, the determinant of A is zero. Therefore, A is singular. This satisfies statement (II).
Since A is singular, the equation Av = 0 has a nontrivial solution. Therefore, 0 is an eigenvalue of A. This satisfies statement (III).
However, we cannot conclude that A is non-singular based on the given information. It is possible for a matrix to be singular even if the sum of all the entries in each row is zero. Therefore, statement (I) is not true.
Hence, the correct answer is option (c): (II) and (III) are true.
To determine whether the given matrix A is non-singular or singular and whether 0 is an eigenvalue of A, we can use the properties of the matrix and its row sums.
Properties of A:
1. A is an n x n matrix with real entries.
2. The sum of all the entries in each row is zero.
Statement (I): A is non-singular
To determine whether A is non-singular, we need to check if the determinant of A is nonzero. If the determinant is nonzero, then A is non-singular. Otherwise, A is singular.
Statement (II): A is singular
To determine whether A is singular, we need to check if the determinant of A is zero. If the determinant is zero, then A is singular. Otherwise, A is non-singular.
Statement (III): 0 is an eigenvalue of A
To determine whether 0 is an eigenvalue of A, we need to check if the equation Av = 0 has a nontrivial solution. If the equation has a nontrivial solution, then 0 is an eigenvalue of A. Otherwise, 0 is not an eigenvalue of A.
Explanation:
Since the sum of all the entries in each row of A is zero, we can say that the rows of A are linearly dependent. This implies that the rank of A is less than n.
Since the rank of A is less than n, the determinant of A is zero. Therefore, A is singular. This satisfies statement (II).
Since A is singular, the equation Av = 0 has a nontrivial solution. Therefore, 0 is an eigenvalue of A. This satisfies statement (III).
However, we cannot conclude that A is non-singular based on the given information. It is possible for a matrix to be singular even if the sum of all the entries in each row is zero. Therefore, statement (I) is not true.
Hence, the correct answer is option (c): (II) and (III) are true.
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Let A = [ajj] be an n x n matrix with real entries such that the sum of all the entries in each row is zero. Consider the following statements(I) A is non-singular(II) A is singular(III) 0 is an eigenvalue of AWhich of the following is correct?a)Only (I) is trueb)(I) and (III) are truec)(II) and (III) are trued)Only (III) is trueCorrect answer is option 'C'. Can you explain this answer?
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