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If the rank of a matrix A is 2, then the rank of 2A is
- a)4
- b)1
- c)2
- d)none of these
Correct answer is option 'C'. Can you explain this answer?
Most Upvoted Answer
If the rank of a matrix A is 2, then the rank of 2A isa)4b)1c)2d)none ...
Explanation:
Rank of a matrix represents the number of linearly independent rows or columns in the matrix. If the rank of a matrix A is 2, then it means that there are 2 linearly independent rows or columns in the matrix.
To find the rank of 2A, we need to consider the matrix 2A and perform row or column operations to bring it into row or column echelon form.
Let A be an m x n matrix, then 2A will also be an m x n matrix.
Suppose we perform row or column operations on 2A to bring it into row echelon form. Then the first row or column will be non-zero as the first row or column of A is non-zero. Therefore, the rank of 2A will be at least 1.
Now, let's consider the case where A is a 2 x 2 matrix with rank 2. Then, the two rows or columns of A are linearly independent. Let A = [a1 a2] where a1 and a2 are the two columns of A.
If we multiply A by 2, we get 2A = [2a1 2a2]. Since a1 and a2 are linearly independent, 2a1 and 2a2 are also linearly independent. Therefore, the rank of 2A is also 2.
Hence, the correct answer is option C.
Rank of a matrix represents the number of linearly independent rows or columns in the matrix. If the rank of a matrix A is 2, then it means that there are 2 linearly independent rows or columns in the matrix.
To find the rank of 2A, we need to consider the matrix 2A and perform row or column operations to bring it into row or column echelon form.
Let A be an m x n matrix, then 2A will also be an m x n matrix.
Suppose we perform row or column operations on 2A to bring it into row echelon form. Then the first row or column will be non-zero as the first row or column of A is non-zero. Therefore, the rank of 2A will be at least 1.
Now, let's consider the case where A is a 2 x 2 matrix with rank 2. Then, the two rows or columns of A are linearly independent. Let A = [a1 a2] where a1 and a2 are the two columns of A.
If we multiply A by 2, we get 2A = [2a1 2a2]. Since a1 and a2 are linearly independent, 2a1 and 2a2 are also linearly independent. Therefore, the rank of 2A is also 2.
Hence, the correct answer is option C.
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If the rank of a matrix A is 2, then the rank of 2A isa)4b)1c)2d)none ...
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If the rank of a matrix A is 2, then the rank of 2A isa)4b)1c)2d)none of theseCorrect answer is option 'C'. Can you explain this answer? for IIT JAM 2025 is part of IIT JAM preparation. The Question and answers have been prepared according to the IIT JAM exam syllabus. Information about If the rank of a matrix A is 2, then the rank of 2A isa)4b)1c)2d)none of theseCorrect answer is option 'C'. Can you explain this answer? covers all topics & solutions for IIT JAM 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for If the rank of a matrix A is 2, then the rank of 2A isa)4b)1c)2d)none of theseCorrect answer is option 'C'. Can you explain this answer?.
If the rank of a matrix A is 2, then the rank of 2A isa)4b)1c)2d)none of theseCorrect answer is option 'C'. Can you explain this answer? for IIT JAM 2025 is part of IIT JAM preparation. The Question and answers have been prepared according to the IIT JAM exam syllabus. Information about If the rank of a matrix A is 2, then the rank of 2A isa)4b)1c)2d)none of theseCorrect answer is option 'C'. Can you explain this answer? covers all topics & solutions for IIT JAM 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for If the rank of a matrix A is 2, then the rank of 2A isa)4b)1c)2d)none of theseCorrect answer is option 'C'. Can you explain this answer?.
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