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Study the following assertions about a square matrix
(i) The sum of the eigen values of A is equal to its trace
(ii) The product of the eigen values of A is equal to its determinant
(iii) All eigen values of A are non-zero, if and only if A is non-singular
(iv) If A-1 exists, then the eigen-values of A-1 are equal to the reciprocal of the eigen
(i) The sum of the eigen values of A is equal to its trace
(ii) The product of the eigen values of A is equal to its determinant
(iii) All eigen values of A are non-zero, if and only if A is non-singular
(iv) If A-1 exists, then the eigen-values of A-1 are equal to the reciprocal of the eigen
Q. Which of the following is correct with respect to above assertions?
- a)Only (iii) and (iv) are true
- b)Only (i) and (ii) are true
- c)Only (ii), (iii) and (iv) are true
- d)(i), (ii), (iii), (iv) all are true
Correct answer is option 'D'. Can you explain this answer?
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Study the following assertions about a square matrix(i) The sum of the...
Assertion about Square Matrix
Sum of Eigen Values
The sum of eigenvalues of a square matrix is equal to the trace of the matrix.
Product of Eigen Values
The product of eigenvalues of a square matrix is equal to the determinant of the matrix.
Non-zero Eigen Values
All eigenvalues of a square matrix are non-zero, if and only if the matrix is non-singular.
Eigen Values of Inverse Matrix
If A-1 exists, then the eigenvalues of A-1 are equal to the reciprocal of the eigenvalues of A.
Correct Assertion
All the above assertions are correct.
Explanation
- The sum of eigenvalues of a matrix is equal to the trace of a matrix. This is because the trace of a matrix is equal to the sum of its diagonal elements, which are also the eigenvalues of the matrix.
- The product of eigenvalues of a matrix is equal to the determinant of a matrix. This is because the determinant of a matrix is equal to the product of its eigenvalues.
- All eigenvalues of a matrix are non-zero if and only if the matrix is non-singular. This is because a matrix is singular if and only if its determinant is zero, which means at least one of its eigenvalues is zero.
- If A-1 exists, then the eigenvalues of A-1 are equal to the reciprocal of the eigenvalues of A. This is because the eigenvalues of A-1 are the inverse of the eigenvalues of A.
Hence, all the given assertions are correct and option D is the correct answer.
Sum of Eigen Values
The sum of eigenvalues of a square matrix is equal to the trace of the matrix.
Product of Eigen Values
The product of eigenvalues of a square matrix is equal to the determinant of the matrix.
Non-zero Eigen Values
All eigenvalues of a square matrix are non-zero, if and only if the matrix is non-singular.
Eigen Values of Inverse Matrix
If A-1 exists, then the eigenvalues of A-1 are equal to the reciprocal of the eigenvalues of A.
Correct Assertion
All the above assertions are correct.
Explanation
- The sum of eigenvalues of a matrix is equal to the trace of a matrix. This is because the trace of a matrix is equal to the sum of its diagonal elements, which are also the eigenvalues of the matrix.
- The product of eigenvalues of a matrix is equal to the determinant of a matrix. This is because the determinant of a matrix is equal to the product of its eigenvalues.
- All eigenvalues of a matrix are non-zero if and only if the matrix is non-singular. This is because a matrix is singular if and only if its determinant is zero, which means at least one of its eigenvalues is zero.
- If A-1 exists, then the eigenvalues of A-1 are equal to the reciprocal of the eigenvalues of A. This is because the eigenvalues of A-1 are the inverse of the eigenvalues of A.
Hence, all the given assertions are correct and option D is the correct answer.
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Study the following assertions about a square matrix(i) The sum of the eigen values of A is equal to its trace(ii) The product of the eigen values of A is equal to its determinant(iii) All eigen values of A are non-zero, if and only if A is non-singular(iv) If A-1 exists, then the eigen-values of A-1 are equal to the reciprocal of the eigenQ. Which of the following is correct with respect to above assertions?a)Only (iii) and (iv) are trueb)Only (i) and (ii) are truec)Only (ii), (iii) and (iv) are trued)(i), (ii), (iii), (iv) all are trueCorrect answer is option 'D'. Can you explain this answer?
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Study the following assertions about a square matrix(i) The sum of the eigen values of A is equal to its trace(ii) The product of the eigen values of A is equal to its determinant(iii) All eigen values of A are non-zero, if and only if A is non-singular(iv) If A-1 exists, then the eigen-values of A-1 are equal to the reciprocal of the eigenQ. Which of the following is correct with respect to above assertions?a)Only (iii) and (iv) are trueb)Only (i) and (ii) are truec)Only (ii), (iii) and (iv) are trued)(i), (ii), (iii), (iv) all are trueCorrect answer is option 'D'. Can you explain this answer? for IIT JAM 2024 is part of IIT JAM preparation. The Question and answers have been prepared according to the IIT JAM exam syllabus. Information about Study the following assertions about a square matrix(i) The sum of the eigen values of A is equal to its trace(ii) The product of the eigen values of A is equal to its determinant(iii) All eigen values of A are non-zero, if and only if A is non-singular(iv) If A-1 exists, then the eigen-values of A-1 are equal to the reciprocal of the eigenQ. Which of the following is correct with respect to above assertions?a)Only (iii) and (iv) are trueb)Only (i) and (ii) are truec)Only (ii), (iii) and (iv) are trued)(i), (ii), (iii), (iv) all are trueCorrect answer is option 'D'. Can you explain this answer? covers all topics & solutions for IIT JAM 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Study the following assertions about a square matrix(i) The sum of the eigen values of A is equal to its trace(ii) The product of the eigen values of A is equal to its determinant(iii) All eigen values of A are non-zero, if and only if A is non-singular(iv) If A-1 exists, then the eigen-values of A-1 are equal to the reciprocal of the eigenQ. Which of the following is correct with respect to above assertions?a)Only (iii) and (iv) are trueb)Only (i) and (ii) are truec)Only (ii), (iii) and (iv) are trued)(i), (ii), (iii), (iv) all are trueCorrect answer is option 'D'. Can you explain this answer?.
Study the following assertions about a square matrix(i) The sum of the eigen values of A is equal to its trace(ii) The product of the eigen values of A is equal to its determinant(iii) All eigen values of A are non-zero, if and only if A is non-singular(iv) If A-1 exists, then the eigen-values of A-1 are equal to the reciprocal of the eigenQ. Which of the following is correct with respect to above assertions?a)Only (iii) and (iv) are trueb)Only (i) and (ii) are truec)Only (ii), (iii) and (iv) are trued)(i), (ii), (iii), (iv) all are trueCorrect answer is option 'D'. Can you explain this answer? for IIT JAM 2024 is part of IIT JAM preparation. The Question and answers have been prepared according to the IIT JAM exam syllabus. Information about Study the following assertions about a square matrix(i) The sum of the eigen values of A is equal to its trace(ii) The product of the eigen values of A is equal to its determinant(iii) All eigen values of A are non-zero, if and only if A is non-singular(iv) If A-1 exists, then the eigen-values of A-1 are equal to the reciprocal of the eigenQ. Which of the following is correct with respect to above assertions?a)Only (iii) and (iv) are trueb)Only (i) and (ii) are truec)Only (ii), (iii) and (iv) are trued)(i), (ii), (iii), (iv) all are trueCorrect answer is option 'D'. Can you explain this answer? covers all topics & solutions for IIT JAM 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Study the following assertions about a square matrix(i) The sum of the eigen values of A is equal to its trace(ii) The product of the eigen values of A is equal to its determinant(iii) All eigen values of A are non-zero, if and only if A is non-singular(iv) If A-1 exists, then the eigen-values of A-1 are equal to the reciprocal of the eigenQ. Which of the following is correct with respect to above assertions?a)Only (iii) and (iv) are trueb)Only (i) and (ii) are truec)Only (ii), (iii) and (iv) are trued)(i), (ii), (iii), (iv) all are trueCorrect answer is option 'D'. Can you explain this answer?.
Solutions for Study the following assertions about a square matrix(i) The sum of the eigen values of A is equal to its trace(ii) The product of the eigen values of A is equal to its determinant(iii) All eigen values of A are non-zero, if and only if A is non-singular(iv) If A-1 exists, then the eigen-values of A-1 are equal to the reciprocal of the eigenQ. Which of the following is correct with respect to above assertions?a)Only (iii) and (iv) are trueb)Only (i) and (ii) are truec)Only (ii), (iii) and (iv) are trued)(i), (ii), (iii), (iv) all are trueCorrect answer is option 'D'. Can you explain this answer? in English & in Hindi are available as part of our courses for IIT JAM.
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Here you can find the meaning of Study the following assertions about a square matrix(i) The sum of the eigen values of A is equal to its trace(ii) The product of the eigen values of A is equal to its determinant(iii) All eigen values of A are non-zero, if and only if A is non-singular(iv) If A-1 exists, then the eigen-values of A-1 are equal to the reciprocal of the eigenQ. Which of the following is correct with respect to above assertions?a)Only (iii) and (iv) are trueb)Only (i) and (ii) are truec)Only (ii), (iii) and (iv) are trued)(i), (ii), (iii), (iv) all are trueCorrect answer is option 'D'. Can you explain this answer? defined & explained in the simplest way possible. Besides giving the explanation of
Study the following assertions about a square matrix(i) The sum of the eigen values of A is equal to its trace(ii) The product of the eigen values of A is equal to its determinant(iii) All eigen values of A are non-zero, if and only if A is non-singular(iv) If A-1 exists, then the eigen-values of A-1 are equal to the reciprocal of the eigenQ. Which of the following is correct with respect to above assertions?a)Only (iii) and (iv) are trueb)Only (i) and (ii) are truec)Only (ii), (iii) and (iv) are trued)(i), (ii), (iii), (iv) all are trueCorrect answer is option 'D'. Can you explain this answer?, a detailed solution for Study the following assertions about a square matrix(i) The sum of the eigen values of A is equal to its trace(ii) The product of the eigen values of A is equal to its determinant(iii) All eigen values of A are non-zero, if and only if A is non-singular(iv) If A-1 exists, then the eigen-values of A-1 are equal to the reciprocal of the eigenQ. Which of the following is correct with respect to above assertions?a)Only (iii) and (iv) are trueb)Only (i) and (ii) are truec)Only (ii), (iii) and (iv) are trued)(i), (ii), (iii), (iv) all are trueCorrect answer is option 'D'. Can you explain this answer? has been provided alongside types of Study the following assertions about a square matrix(i) The sum of the eigen values of A is equal to its trace(ii) The product of the eigen values of A is equal to its determinant(iii) All eigen values of A are non-zero, if and only if A is non-singular(iv) If A-1 exists, then the eigen-values of A-1 are equal to the reciprocal of the eigenQ. Which of the following is correct with respect to above assertions?a)Only (iii) and (iv) are trueb)Only (i) and (ii) are truec)Only (ii), (iii) and (iv) are trued)(i), (ii), (iii), (iv) all are trueCorrect answer is option 'D'. Can you explain this answer? theory, EduRev gives you an
ample number of questions to practice Study the following assertions about a square matrix(i) The sum of the eigen values of A is equal to its trace(ii) The product of the eigen values of A is equal to its determinant(iii) All eigen values of A are non-zero, if and only if A is non-singular(iv) If A-1 exists, then the eigen-values of A-1 are equal to the reciprocal of the eigenQ. Which of the following is correct with respect to above assertions?a)Only (iii) and (iv) are trueb)Only (i) and (ii) are truec)Only (ii), (iii) and (iv) are trued)(i), (ii), (iii), (iv) all are trueCorrect answer is option 'D'. Can you explain this answer? tests, examples and also practice IIT JAM tests.
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