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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)
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?
Most Upvoted Answer
Let A and B two n × n matrices over real numbers. Let rank(M) an...
Concept:
Properties of Rank: Rank of a matrix is the number of independent rows in the given matrix. Given two square matrices A and B of order n × n, we have following properties:
- 1. Rank of product of A and B i.e. Rank (AB) ≥ Rank (A) + Rank (B) – order of square matrix
- 2. Rank of sum of A and B i.e. Rank (A + B) ≤ Rank (A) + Rank (B).
Properties of Determinant: Given two square matrices A and B of order n × n and their determinants Det(A) and Det(B) respectively, determinant of their product i.e Det( AB) = Det(A) * Det(B). However, the same does not hold for the addition of the given matrices.
Example:
Consider two square matrices A and B each of order 2×2.
Rank of B = 2
Det (B) = 1
Rank of B = 2
Det (B) = 1
Rank (AB) = 2
Det (AB) = 5
Rank of A+B = 2
Det (A+B) = 12
Statement I is FALSE.
The rank of product matrix AB is 2. Product of Rank(A) and Rank(B) is: 2*2 = 4. Therefore, the rank of the product matrix is not equal to the product of the rank of individual matrices.
The rank of product matrix AB is 2. Product of Rank(A) and Rank(B) is: 2*2 = 4. Therefore, the rank of the product matrix is not equal to the product of the rank of individual matrices.
Statement II is TRUE.
Det(AB)= 5 = Det(A) * Det(B)
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.
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.
Statement IV is FALSE.
Det(A+B)= 12, which is greater than the sum of the determinants of individual matrices.
Det(A+B)= 12, which is greater than the sum of the determinants of individual matrices.
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Let A and B two n × n matrices over real numbers. Let rank(M) an...
Explanation:
Rank and Determinant Properties:
- The rank of a matrix is the maximum number of linearly independent rows or columns in the matrix.
- The determinant of a matrix is a scalar value that can be computed from its elements.
Analysis of the Statements:
I. rank(AB) = rank(A) rank(B)
- This statement is NOT true in general. The rank of the product of two matrices is at most the minimum of the ranks of the matrices.
II. det(AB) = det(A) det(B)
- This statement is NOT true in general. The determinant of the product of two matrices is equal to the product of their determinants only if the matrices commute.
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.
IV. det(A + B) ≤ det(A) + det(B)
- This statement is NOT true in general. The determinant of the sum of two matrices is not necessarily less than or equal to the sum of their determinants.
Therefore, the correct answer is option C (II and III only), as statement III is true and statement II is false.
Rank and Determinant Properties:
- The rank of a matrix is the maximum number of linearly independent rows or columns in the matrix.
- The determinant of a matrix is a scalar value that can be computed from its elements.
Analysis of the Statements:
I. rank(AB) = rank(A) rank(B)
- This statement is NOT true in general. The rank of the product of two matrices is at most the minimum of the ranks of the matrices.
II. det(AB) = det(A) det(B)
- This statement is NOT true in general. The determinant of the product of two matrices is equal to the product of their determinants only if the matrices commute.
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.
IV. det(A + B) ≤ det(A) + det(B)
- This statement is NOT true in general. The determinant of the sum of two matrices is not necessarily less than or equal to the sum of their determinants.
Therefore, the correct answer is option C (II and III only), as statement III is true and statement II is false.
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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?, 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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