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If A and B are 3 × 3 real matrices such that rank (AB) = 1, then rank (BA) cannot be
- a)3
- b)0
- c)2
- d)1
Correct answer is option 'A'. Can you explain this answer?
Verified Answer
If A and B are 3 × 3 real matrices such that rank (AB) = 1, then...
Here A & B a re 3 × 2 real matrices such that rank (AB) = 1
So, |AB| = 0 ⇒ |A| |B| = 0 (∴ |AB| = |A| |B|)
⇒ either |A| or |B| should be zero
So, |BA| = |B||A| = 0
⇒ BA is singular
Hence rank (BA) cannot be 3. (Because BA is 3 × 3 matrix)
So, |AB| = 0 ⇒ |A| |B| = 0 (∴ |AB| = |A| |B|)
⇒ either |A| or |B| should be zero
So, |BA| = |B||A| = 0
⇒ BA is singular
Hence rank (BA) cannot be 3. (Because BA is 3 × 3 matrix)
Most Upvoted Answer
If A and B are 3 × 3 real matrices such that rank (AB) = 1, then...
Here A & B a re 3 × 2 real matrices such that rank (AB) = 1
So, |AB| = 0 ⇒ |A| |B| = 0 (∴ |AB| = |A| |B|)
⇒ either |A| or |B| should be zero
So, |BA| = |B||A| = 0
⇒ BA is singular
Hence rank (BA) cannot be 3. (Because BA is 3 × 3 matrix)
So, |AB| = 0 ⇒ |A| |B| = 0 (∴ |AB| = |A| |B|)
⇒ either |A| or |B| should be zero
So, |BA| = |B||A| = 0
⇒ BA is singular
Hence rank (BA) cannot be 3. (Because BA is 3 × 3 matrix)
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If A and B are 3 × 3 real matrices such that rank (AB) = 1, then...
Understanding Matrix Ranks
When dealing with the rank of matrices, particularly the product of two matrices, it is essential to recognize how the ranks of individual matrices affect the rank of their products.
Given Conditions
- Let A and B be 3 × 3 real matrices.
- We know that rank(AB) = 1.
Implications of rank(AB) = 1
- The rank of a product of matrices cannot exceed the rank of either matrix. Thus, we have:
- rank(AB) ≤ min(rank(A), rank(B)).
- Since rank(AB) = 1, it follows that both rank(A) and rank(B) must be at least 1.
Analyzing rank(BA)
- We want to find the possible ranks of BA. The rank-nullity theorem tells us that:
- rank(AB) + nullity(B) = number of columns of B = 3.
- Since rank(AB) = 1, it implies nullity(B) = 2. Thus, B has a rank of 1.
Possible Ranks for A
- rank(A) can be either 1, 2, or 3. However, since B has rank 1, the rank of the product BA can be analyzed:
- If rank(A) = 3, then rank(BA) would be at least 1, but it cannot be 3 since B is of rank 1.
- If rank(A) = 2, then again rank(BA) cannot be 2, as B can only contribute a maximum rank of 1.
- If rank(A) = 1, then rank(BA) = 1.
Conclusion
- Thus, the only possible ranks for BA under the given conditions are 0 or 1. Therefore, the maximum rank for BA cannot be 3, making option 'A' the correct answer.
When dealing with the rank of matrices, particularly the product of two matrices, it is essential to recognize how the ranks of individual matrices affect the rank of their products.
Given Conditions
- Let A and B be 3 × 3 real matrices.
- We know that rank(AB) = 1.
Implications of rank(AB) = 1
- The rank of a product of matrices cannot exceed the rank of either matrix. Thus, we have:
- rank(AB) ≤ min(rank(A), rank(B)).
- Since rank(AB) = 1, it follows that both rank(A) and rank(B) must be at least 1.
Analyzing rank(BA)
- We want to find the possible ranks of BA. The rank-nullity theorem tells us that:
- rank(AB) + nullity(B) = number of columns of B = 3.
- Since rank(AB) = 1, it implies nullity(B) = 2. Thus, B has a rank of 1.
Possible Ranks for A
- rank(A) can be either 1, 2, or 3. However, since B has rank 1, the rank of the product BA can be analyzed:
- If rank(A) = 3, then rank(BA) would be at least 1, but it cannot be 3 since B is of rank 1.
- If rank(A) = 2, then again rank(BA) cannot be 2, as B can only contribute a maximum rank of 1.
- If rank(A) = 1, then rank(BA) = 1.
Conclusion
- Thus, the only possible ranks for BA under the given conditions are 0 or 1. Therefore, the maximum rank for BA cannot be 3, making option 'A' the correct answer.
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