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Let A and B two n × n matrices over real numbers. Let rank(M) and det(M) denote the rank and determinant of a matrix M, respectively. Consider the following statements.
I. rank(AB) = rank(A) rank(B)
II. det(AB) = det(A) det(B)
III. rank(A + B) ≤ rank(A) + rank(B)
IV. det(A + B) ≤ det(A) + det(B)
Which of the above statements are TRUE?
- a)I and II only
- b)I and IV only
- c)II and III only
- d)III and IV only
Correct answer is option 'C'. Can you explain this answer?
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Most Upvoted Answer
Let A and B two n × n matrices over real numbers. Let rank(M) an...
Understanding the Statements
In the context of two \( n \times n \) matrices \( A \) and \( B \), we need to evaluate the truth of the given statements.
Statement I: rank(AB) = rank(A) rank(B)
- This statement is generally false. The rank of the product \( AB \) can be less than or equal to the minimum of the ranks of \( A \) and \( B \). It does not hold that \( \text{rank}(AB) = \text{rank}(A) \cdot \text{rank}(B) \).
Statement II: det(AB) = det(A) det(B)
- This statement is true. The property of determinants states that the determinant of a product of matrices is equal to the product of their determinants, i.e., \( \det(AB) = \det(A) \cdot \det(B) \).
Statement III: rank(A + B) ≤ rank(A) + rank(B)
- This statement is true. The rank of the sum of two matrices is at most the sum of their ranks. This is a well-known property in linear algebra.
Statement IV: det(A + B) ≤ det(A) + det(B)
- This statement is false in general. The determinant of the sum of two matrices does not necessarily satisfy this inequality, particularly for cases where \( A \) and \( B \) are not simultaneously diagonalizable.
Conclusion
- The true statements are II and III. Therefore, the correct answer is option 'C': II and III only.
In the context of two \( n \times n \) matrices \( A \) and \( B \), we need to evaluate the truth of the given statements.
Statement I: rank(AB) = rank(A) rank(B)
- This statement is generally false. The rank of the product \( AB \) can be less than or equal to the minimum of the ranks of \( A \) and \( B \). It does not hold that \( \text{rank}(AB) = \text{rank}(A) \cdot \text{rank}(B) \).
Statement II: det(AB) = det(A) det(B)
- This statement is true. The property of determinants states that the determinant of a product of matrices is equal to the product of their determinants, i.e., \( \det(AB) = \det(A) \cdot \det(B) \).
Statement III: rank(A + B) ≤ rank(A) + rank(B)
- This statement is true. The rank of the sum of two matrices is at most the sum of their ranks. This is a well-known property in linear algebra.
Statement IV: det(A + B) ≤ det(A) + det(B)
- This statement is false in general. The determinant of the sum of two matrices does not necessarily satisfy this inequality, particularly for cases where \( A \) and \( B \) are not simultaneously diagonalizable.
Conclusion
- The true statements are II and III. Therefore, the correct answer is option 'C': II and III only.
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Let A and B two n × n matrices over real numbers. Let rank(M) an...
Example:
Consider two square matrices A and B each of order 2×2.
Consider two square matrices A and B each of order 2×2.
Statement II is TRUE.
Det(AB)= 5 = Det(A) * Det(B)
Statement III is TRUE.
Rank(A + B) = 2. Sum of rank of A and B is: 2 + 2 = 4. Therefore, the relation: Rank (A + B) ≤ Rank (A) + Rank (B) holds true
Therefore, the rank of the addition matrix is less than or equal to the sum of the rank of the individual matrices.
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Let A and B two n × n matrices over real numbers. Let rank(M) and det(M) denote the rank and determinant of a matrix M, respectively. Consider the following statements.I. rank(AB) = rank(A) rank(B)II. det(AB) = det(A) det(B)III. rank(A + B) ≤ rank(A) + rank(B)IV. det(A + B) ≤ det(A) + det(B)Which of the above statements are TRUE?a)I and II onlyb)I and IV onlyc)II and III onlyd)III and IV onlyCorrect answer is option 'C'. Can you explain this answer?
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Let A and B two n × n matrices over real numbers. Let rank(M) and det(M) denote the rank and determinant of a matrix M, respectively. Consider the following statements.I. rank(AB) = rank(A) rank(B)II. det(AB) = det(A) det(B)III. rank(A + B) ≤ rank(A) + rank(B)IV. det(A + B) ≤ det(A) + det(B)Which of the above statements are TRUE?a)I and II onlyb)I and IV onlyc)II and III onlyd)III and IV onlyCorrect answer is option 'C'. Can you explain this answer? for Mathematics 2024 is part of Mathematics preparation. The Question and answers have been prepared according to the Mathematics exam syllabus. Information about Let A and B two n × n matrices over real numbers. Let rank(M) and det(M) denote the rank and determinant of a matrix M, respectively. Consider the following statements.I. rank(AB) = rank(A) rank(B)II. det(AB) = det(A) det(B)III. rank(A + B) ≤ rank(A) + rank(B)IV. det(A + B) ≤ det(A) + det(B)Which of the above statements are TRUE?a)I and II onlyb)I and IV onlyc)II and III onlyd)III and IV onlyCorrect answer is option 'C'. Can you explain this answer? covers all topics & solutions for Mathematics 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Let A and B two n × n matrices over real numbers. Let rank(M) and det(M) denote the rank and determinant of a matrix M, respectively. Consider the following statements.I. rank(AB) = rank(A) rank(B)II. det(AB) = det(A) det(B)III. rank(A + B) ≤ rank(A) + rank(B)IV. det(A + B) ≤ det(A) + det(B)Which of the above statements are TRUE?a)I and II onlyb)I and IV onlyc)II and III onlyd)III and IV onlyCorrect answer is option 'C'. Can you explain this answer?.
Let A and B two n × n matrices over real numbers. Let rank(M) and det(M) denote the rank and determinant of a matrix M, respectively. Consider the following statements.I. rank(AB) = rank(A) rank(B)II. det(AB) = det(A) det(B)III. rank(A + B) ≤ rank(A) + rank(B)IV. det(A + B) ≤ det(A) + det(B)Which of the above statements are TRUE?a)I and II onlyb)I and IV onlyc)II and III onlyd)III and IV onlyCorrect answer is option 'C'. Can you explain this answer? for Mathematics 2024 is part of Mathematics preparation. The Question and answers have been prepared according to the Mathematics exam syllabus. Information about Let A and B two n × n matrices over real numbers. Let rank(M) and det(M) denote the rank and determinant of a matrix M, respectively. Consider the following statements.I. rank(AB) = rank(A) rank(B)II. det(AB) = det(A) det(B)III. rank(A + B) ≤ rank(A) + rank(B)IV. det(A + B) ≤ det(A) + det(B)Which of the above statements are TRUE?a)I and II onlyb)I and IV onlyc)II and III onlyd)III and IV onlyCorrect answer is option 'C'. Can you explain this answer? covers all topics & solutions for Mathematics 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Let A and B two n × n matrices over real numbers. Let rank(M) and det(M) denote the rank and determinant of a matrix M, respectively. Consider the following statements.I. rank(AB) = rank(A) rank(B)II. det(AB) = det(A) det(B)III. rank(A + B) ≤ rank(A) + rank(B)IV. det(A + B) ≤ det(A) + det(B)Which of the above statements are TRUE?a)I and II onlyb)I and IV onlyc)II and III onlyd)III and IV onlyCorrect answer is option 'C'. Can you explain this answer?.
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Let A and B two n × n matrices over real numbers. Let rank(M) and det(M) denote the rank and determinant of a matrix M, respectively. Consider the following statements.I. rank(AB) = rank(A) rank(B)II. det(AB) = det(A) det(B)III. rank(A + B) ≤ rank(A) + rank(B)IV. det(A + B) ≤ det(A) + det(B)Which of the above statements are TRUE?a)I and II onlyb)I and IV onlyc)II and III onlyd)III and IV onlyCorrect answer is option 'C'. Can you explain this answer?, a detailed solution for Let A and B two n × n matrices over real numbers. Let rank(M) and det(M) denote the rank and determinant of a matrix M, respectively. Consider the following statements.I. rank(AB) = rank(A) rank(B)II. det(AB) = det(A) det(B)III. rank(A + B) ≤ rank(A) + rank(B)IV. det(A + B) ≤ det(A) + det(B)Which of the above statements are TRUE?a)I and II onlyb)I and IV onlyc)II and III onlyd)III and IV onlyCorrect answer is option 'C'. Can you explain this answer? has been provided alongside types of Let A and B two n × n matrices over real numbers. Let rank(M) and det(M) denote the rank and determinant of a matrix M, respectively. Consider the following statements.I. rank(AB) = rank(A) rank(B)II. det(AB) = det(A) det(B)III. rank(A + B) ≤ rank(A) + rank(B)IV. det(A + B) ≤ det(A) + det(B)Which of the above statements are TRUE?a)I and II onlyb)I and IV onlyc)II and III onlyd)III and IV onlyCorrect answer is option 'C'. Can you explain this answer? theory, EduRev gives you an
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