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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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