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If A is singular matrix of order n, then a (adj A) equals,
- a)null matrix
- b)row matrix
- c)identity matrix
- d)none of these
Correct answer is option 'A'. Can you explain this answer?
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Most Upvoted Answer
If A is singular matrix of order n, then a (adj A) equals,a)null matri...
Explanation:
To understand why the product of a scalar 'a' and the adjoint of a singular matrix A is the null matrix, we need to review some key concepts.
Definition:
A matrix A is said to be singular if its determinant (denoted by |A|) is equal to zero.
Adjoint of a Matrix:
The adjoint of a matrix A, denoted by adj(A), is the transpose of the matrix of cofactors of A. Each element in the adjoint matrix is obtained by taking the cofactor of the corresponding element in the original matrix and then transposing it.
Properties of Adjoint Matrix:
1. For any square matrix A, adj(adj(A)) = |A|^(n-2) * A, where n is the order of the matrix A.
2. If A is a singular matrix, then |A| = 0. Therefore, adj(A) * A = |A|^(n-2) * A = 0^(n-2) * A = 0 * A = 0 matrix.
Proof:
Let A be a singular matrix of order n. Since A is singular, |A| = 0.
From the properties of the adjoint matrix, we have:
adj(A) * A = |A|^(n-2) * A
Substituting |A| = 0, we get:
adj(A) * A = 0^(n-2) * A = 0 * A = 0 matrix
Therefore, the product of a scalar 'a' and the adjoint of a singular matrix A is the null matrix.
Conclusion:
The correct answer is option 'A': null matrix.
To understand why the product of a scalar 'a' and the adjoint of a singular matrix A is the null matrix, we need to review some key concepts.
Definition:
A matrix A is said to be singular if its determinant (denoted by |A|) is equal to zero.
Adjoint of a Matrix:
The adjoint of a matrix A, denoted by adj(A), is the transpose of the matrix of cofactors of A. Each element in the adjoint matrix is obtained by taking the cofactor of the corresponding element in the original matrix and then transposing it.
Properties of Adjoint Matrix:
1. For any square matrix A, adj(adj(A)) = |A|^(n-2) * A, where n is the order of the matrix A.
2. If A is a singular matrix, then |A| = 0. Therefore, adj(A) * A = |A|^(n-2) * A = 0^(n-2) * A = 0 * A = 0 matrix.
Proof:
Let A be a singular matrix of order n. Since A is singular, |A| = 0.
From the properties of the adjoint matrix, we have:
adj(A) * A = |A|^(n-2) * A
Substituting |A| = 0, we get:
adj(A) * A = 0^(n-2) * A = 0 * A = 0 matrix
Therefore, the product of a scalar 'a' and the adjoint of a singular matrix A is the null matrix.
Conclusion:
The correct answer is option 'A': null matrix.
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If A is singular matrix of order n, then a (adj A) equals,a)null matrixb)row matrixc)identity matrixd)none of theseCorrect answer is option 'A'. Can you explain this answer?
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If A is singular matrix of order n, then a (adj A) equals,a)null matrixb)row matrixc)identity matrixd)none of theseCorrect answer is option 'A'. Can you explain this answer? for JEE 2024 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about If A is singular matrix of order n, then a (adj A) equals,a)null matrixb)row matrixc)identity matrixd)none of theseCorrect answer is option 'A'. Can you explain this answer? covers all topics & solutions for JEE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for If A is singular matrix of order n, then a (adj A) equals,a)null matrixb)row matrixc)identity matrixd)none of theseCorrect answer is option 'A'. Can you explain this answer?.
If A is singular matrix of order n, then a (adj A) equals,a)null matrixb)row matrixc)identity matrixd)none of theseCorrect answer is option 'A'. Can you explain this answer? for JEE 2024 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about If A is singular matrix of order n, then a (adj A) equals,a)null matrixb)row matrixc)identity matrixd)none of theseCorrect answer is option 'A'. Can you explain this answer? covers all topics & solutions for JEE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for If A is singular matrix of order n, then a (adj A) equals,a)null matrixb)row matrixc)identity matrixd)none of theseCorrect answer is option 'A'. Can you explain this answer?.
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