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Let G denotes the group of all 2 x 2 invertible matrices with entries from R. Let H1 ={ A ∈ G: det (A )= 1} and H2= { A ∈ G A is upper triangular} consider the following statement:
P = H1ΔG
Q = H2ΔG, Then,
  • a)
    Both P and Q are true
  • b)
    P is true but Q is false
  • c)
    P is false but Q is true
  • d)
    Both P and Q are false.
Correct answer is option 'A'. Can you explain this answer?
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Let G denotes the group of all 2 x 2 invertible matrices with entries ...

 

Similarly, we can show that 
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Let G denotes the group of all 2 x 2 invertible matrices with entries ...

 

Similarly, we can show that 
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Let G denotes the group of all 2 x 2 invertible matrices with entries ...
To show that H1 is a subgroup of G, we need to show that it satisfies the three conditions for a subgroup:

1. Closure: For any two matrices A and B in H1, their product AB must also be in H1.

Consider two matrices A and B in H1, such that A = [[a, b], [c, d]] and B = [[e, f], [g, h]]. We have:

AB = [[a, b], [c, d]] [[e, f], [g, h]]
= [[ae + bg, af + bh], [ce + dg, cf + dh]]

Since A and B are invertible, their determinants are non-zero, so ad - bc ≠ 0 and eh - fg ≠ 0. Therefore, the determinant of AB is given by:

det(AB) = (ae + bg)(cf + dh) - (af + bh)(ce + dg)
= aecf + aedh + bccf + bcdh - aecf - aedg - bcfg - bcdh
= aedh - aedg - bcfg
= ad(eh - fg)

Since ad - bc ≠ 0, we have det(AB) ≠ 0, which means AB is invertible. Hence, AB ∈ H1, and closure is satisfied.

2. Identity: The identity matrix I = [[1, 0], [0, 1]] is in H1, since its determinant is 1.

3. Inverse: For any matrix A in H1, its inverse A^(-1) = [[d, -b], [-c, a]] is also in H1, since its determinant is (ad - bc)^(-1), which is non-zero.

Therefore, H1 is a subgroup of G.
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Let G denotes the group of all 2 x 2 invertible matrices with entries from R. LetH1 ={ A ∈ G: det (A )= 1} and H2= { A ∈ GA is upper triangular} consider thefollowing statement:P = H1ΔGQ = H2ΔG, Then,a)Both P and Q are trueb)P is true but Q is falsec)P is false but Q is trued)Both P and Q are false.Correct answer is option 'A'. Can you explain this answer?
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Let G denotes the group of all 2 x 2 invertible matrices with entries from R. LetH1 ={ A ∈ G: det (A )= 1} and H2= { A ∈ GA is upper triangular} consider thefollowing statement:P = H1ΔGQ = H2ΔG, Then,a)Both P and Q are trueb)P is true but Q is falsec)P is false but Q is trued)Both P and Q are false.Correct answer is option 'A'. Can you explain this answer? for Mathematics 2024 is part of Mathematics preparation. The Question and answers have been prepared according to the Mathematics exam syllabus. Information about Let G denotes the group of all 2 x 2 invertible matrices with entries from R. LetH1 ={ A ∈ G: det (A )= 1} and H2= { A ∈ GA is upper triangular} consider thefollowing statement:P = H1ΔGQ = H2ΔG, Then,a)Both P and Q are trueb)P is true but Q is falsec)P is false but Q is trued)Both P and Q are false.Correct answer is option 'A'. Can you explain this answer? covers all topics & solutions for Mathematics 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Let G denotes the group of all 2 x 2 invertible matrices with entries from R. LetH1 ={ A ∈ G: det (A )= 1} and H2= { A ∈ GA is upper triangular} consider thefollowing statement:P = H1ΔGQ = H2ΔG, Then,a)Both P and Q are trueb)P is true but Q is falsec)P is false but Q is trued)Both P and Q are false.Correct answer is option 'A'. Can you explain this answer?.
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