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If A is any m×n matrix such that AB and BA are both defined, then B is a matrix of type
- a)m×n
- b)n×m
- c)m×m
- d)n×n
Correct answer is option 'B'. Can you explain this answer?
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If A is any m×n matrix such that AB and BA are both defined, then B is...
Understanding Matrix Dimensions
When dealing with matrices A and B, it's crucial to understand the dimensions involved in matrix multiplication.
Matrix A
- A is an m×n matrix.
- This means A has m rows and n columns.
Matrix B
For the products AB and BA to be defined, the dimensions of B must satisfy certain conditions:
1. For AB to be defined:
- B must have n rows (to match the number of columns in A).
- Therefore, B must be of type m×k where k can be any positive integer.
2. For BA to be defined:
- B must also have m columns (to match the number of rows in A).
- Hence, B must be of type p×m where p can be any positive integer.
Conclusion on Matrix B's Dimensions
Combining these two conditions, we deduce that:
- B must be an n×m matrix for the multiplication AB to be defined.
- B must also be an m×n matrix for the multiplication BA to be defined.
Thus, the only consistent dimension for B that satisfies both conditions is:
n×m
This aligns with option 'B' in your question. Therefore, we conclude that:
- B is of type n×m, which allows for both products AB and BA to be defined.
In summary, the dimensions of matrix B are crucial for ensuring that both matrix products are valid, confirming that B must be an n×m matrix.
When dealing with matrices A and B, it's crucial to understand the dimensions involved in matrix multiplication.
Matrix A
- A is an m×n matrix.
- This means A has m rows and n columns.
Matrix B
For the products AB and BA to be defined, the dimensions of B must satisfy certain conditions:
1. For AB to be defined:
- B must have n rows (to match the number of columns in A).
- Therefore, B must be of type m×k where k can be any positive integer.
2. For BA to be defined:
- B must also have m columns (to match the number of rows in A).
- Hence, B must be of type p×m where p can be any positive integer.
Conclusion on Matrix B's Dimensions
Combining these two conditions, we deduce that:
- B must be an n×m matrix for the multiplication AB to be defined.
- B must also be an m×n matrix for the multiplication BA to be defined.
Thus, the only consistent dimension for B that satisfies both conditions is:
n×m
This aligns with option 'B' in your question. Therefore, we conclude that:
- B is of type n×m, which allows for both products AB and BA to be defined.
In summary, the dimensions of matrix B are crucial for ensuring that both matrix products are valid, confirming that B must be an n×m matrix.
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If A is any m×n matrix such that AB and BA are both defined, then B is a matrix of typea)m×nb)n×mc)m×md)n×nCorrect answer is option 'B'. Can you explain this answer?
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If A is any m×n matrix such that AB and BA are both defined, then B is a matrix of typea)m×nb)n×mc)m×md)n×nCorrect answer is option 'B'. Can you explain this answer? for JEE 2025 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about If A is any m×n matrix such that AB and BA are both defined, then B is a matrix of typea)m×nb)n×mc)m×md)n×nCorrect answer is option 'B'. Can you explain this answer? covers all topics & solutions for JEE 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for If A is any m×n matrix such that AB and BA are both defined, then B is a matrix of typea)m×nb)n×mc)m×md)n×nCorrect answer is option 'B'. Can you explain this answer?.
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