IIT JAM Exam > IIT JAM Questions > Let A be a 3 x 4 matrix of rank 2. Then the r...
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Let A be a 3 x 4 matrix of rank 2. Then the rank of ATA, where AT denotes the transpose of A. is,
- a)exactly 2
- b)exactly 1
- c)at most 3, and atleast 2
- d)at most 2, but not necessarily 2
Correct answer is option 'A'. Can you explain this answer?
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Verified Answer
Let A be a 3 x 4 matrix of rank 2. Then the rank of ATA, where AT deno...
We know Rank A = Rank AT= 2
and also we know Rank (AB) £ min {Rank A, Rank B} as here we are taking product of matrix with its transpose.
Thus Rank (ATA) = 2
and also we know Rank (AB) £ min {Rank A, Rank B} as here we are taking product of matrix with its transpose.
Thus Rank (ATA) = 2
Most Upvoted Answer
Let A be a 3 x 4 matrix of rank 2. Then the rank of ATA, where AT deno...
Rank of a matrix:
The rank of a matrix is the maximum number of linearly independent rows or columns in that matrix. It can be calculated by performing row operations and reducing the matrix to its row echelon form or reduced row echelon form.
Rank of a product of matrices:
For two matrices A and B, the rank of their product AB is at most the minimum of the ranks of A and B. In other words, rank(AB) ≤ min(rank(A), rank(B)).
Rank of a transpose:
The rank of a matrix and its transpose are always the same. So, rank(A) = rank(A^T).
Rank of ATA:
Given a matrix A of rank 2, we need to determine the rank of ATA.
- Since A is a 3 x 4 matrix of rank 2, it means that there are 2 linearly independent rows or columns in A.
- Transposing A will give us a 4 x 3 matrix of rank 2, denoted by A^T.
- Multiplying A and A^T will give us a 3 x 3 matrix, denoted by ATA.
- According to the rank property mentioned earlier, rank(ATA) ≤ min(rank(A), rank(A^T)).
- Since rank(A) = rank(A^T), we can say that rank(ATA) ≤ rank(A).
- But we know that rank(A) = 2, so rank(ATA) ≤ 2.
Conclusion:
From the above analysis, we can conclude that the rank of ATA is at most 2. But since A is a 3 x 4 matrix of rank 2, we have 2 linearly independent rows or columns in A. When we multiply A and A^T to get ATA, we are essentially taking the dot product of rows and columns of A. Since we have 2 linearly independent rows or columns, the dot product will also be linearly independent. Therefore, the rank of ATA is exactly 2.
Hence, the correct answer is option 'A' - exactly 2.
The rank of a matrix is the maximum number of linearly independent rows or columns in that matrix. It can be calculated by performing row operations and reducing the matrix to its row echelon form or reduced row echelon form.
Rank of a product of matrices:
For two matrices A and B, the rank of their product AB is at most the minimum of the ranks of A and B. In other words, rank(AB) ≤ min(rank(A), rank(B)).
Rank of a transpose:
The rank of a matrix and its transpose are always the same. So, rank(A) = rank(A^T).
Rank of ATA:
Given a matrix A of rank 2, we need to determine the rank of ATA.
- Since A is a 3 x 4 matrix of rank 2, it means that there are 2 linearly independent rows or columns in A.
- Transposing A will give us a 4 x 3 matrix of rank 2, denoted by A^T.
- Multiplying A and A^T will give us a 3 x 3 matrix, denoted by ATA.
- According to the rank property mentioned earlier, rank(ATA) ≤ min(rank(A), rank(A^T)).
- Since rank(A) = rank(A^T), we can say that rank(ATA) ≤ rank(A).
- But we know that rank(A) = 2, so rank(ATA) ≤ 2.
Conclusion:
From the above analysis, we can conclude that the rank of ATA is at most 2. But since A is a 3 x 4 matrix of rank 2, we have 2 linearly independent rows or columns in A. When we multiply A and A^T to get ATA, we are essentially taking the dot product of rows and columns of A. Since we have 2 linearly independent rows or columns, the dot product will also be linearly independent. Therefore, the rank of ATA is exactly 2.
Hence, the correct answer is option 'A' - exactly 2.
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Let A be a 3 x 4 matrix of rank 2. Then the rank of ATA, where AT denotes the transpose of A. is,a)exactly 2b)exactly 1c)at most 3, and atleast 2d)at most 2, but not necessarily 2Correct answer is option 'A'. Can you explain this answer?
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Let A be a 3 x 4 matrix of rank 2. Then the rank of ATA, where AT denotes the transpose of A. is,a)exactly 2b)exactly 1c)at most 3, and atleast 2d)at most 2, but not necessarily 2Correct answer is option 'A'. 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 Let A be a 3 x 4 matrix of rank 2. Then the rank of ATA, where AT denotes the transpose of A. is,a)exactly 2b)exactly 1c)at most 3, and atleast 2d)at most 2, but not necessarily 2Correct answer is option 'A'. 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 Let A be a 3 x 4 matrix of rank 2. Then the rank of ATA, where AT denotes the transpose of A. is,a)exactly 2b)exactly 1c)at most 3, and atleast 2d)at most 2, but not necessarily 2Correct answer is option 'A'. Can you explain this answer?.
Let A be a 3 x 4 matrix of rank 2. Then the rank of ATA, where AT denotes the transpose of A. is,a)exactly 2b)exactly 1c)at most 3, and atleast 2d)at most 2, but not necessarily 2Correct answer is option 'A'. 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 Let A be a 3 x 4 matrix of rank 2. Then the rank of ATA, where AT denotes the transpose of A. is,a)exactly 2b)exactly 1c)at most 3, and atleast 2d)at most 2, but not necessarily 2Correct answer is option 'A'. 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 Let A be a 3 x 4 matrix of rank 2. Then the rank of ATA, where AT denotes the transpose of A. is,a)exactly 2b)exactly 1c)at most 3, and atleast 2d)at most 2, but not necessarily 2Correct answer is option 'A'. Can you explain this answer?.
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Let A be a 3 x 4 matrix of rank 2. Then the rank of ATA, where AT denotes the transpose of A. is,a)exactly 2b)exactly 1c)at most 3, and atleast 2d)at most 2, but not necessarily 2Correct answer is option 'A'. Can you explain this answer?, a detailed solution for Let A be a 3 x 4 matrix of rank 2. Then the rank of ATA, where AT denotes the transpose of A. is,a)exactly 2b)exactly 1c)at most 3, and atleast 2d)at most 2, but not necessarily 2Correct answer is option 'A'. Can you explain this answer? has been provided alongside types of Let A be a 3 x 4 matrix of rank 2. Then the rank of ATA, where AT denotes the transpose of A. is,a)exactly 2b)exactly 1c)at most 3, and atleast 2d)at most 2, but not necessarily 2Correct answer is option 'A'. Can you explain this answer? theory, EduRev gives you an
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