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Let P be a 4 × 4 matrix with entries from the set of rational numbers. If 
with 
is a root of the characteristic polynomial of P and I is the 4 × 4 identity matrix, then
with is a root of the characteristic polynomial of P and I is the 4 × 4 identity matrix, then- a)P4 = 2P2 - 9I
- b)P4 = 4P2 - 91
- c)P4 = 4P2 + 9I
- d)P4 = 2P2 + 9I
Correct answer is option 'A'. Can you explain this answer?
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Verified Answer
Let P be a 4 × 4 matrix with entries from the set of rational nu...
Given P is a 4 × 4 matrix with rational entries.
Let characteristic polynomial of P is
hp(x) = x4 + ax3 + bx2 + cx + d, ...(i)
where a, b, c and d are rational.
Since √2 + i is a root of (1), Then √2 − i is also a root of (1)
⇒ (x2 − 22x + 3) is a factor of (1)
Since x2 + 3 − 2√2x is factor of (1), Then
x2 + 3 + 2√2x is also a factor of (1)
⇒ (x2 + 3 − 2√2x) (x2 + 3 + 2√2x) is a factor of (1)
x4 + 6x2 + 9 – 8x2.
= x4 – 2x2 + 9 ...(ii)
From (1) & (2), we have
P4 – 2P2 + 9I = 0
⇒ P4 = 2P2 – 9I
Let characteristic polynomial of P is
hp(x) = x4 + ax3 + bx2 + cx + d, ...(i)
where a, b, c and d are rational.
Since √2 + i is a root of (1), Then √2 − i is also a root of (1)
⇒ (x2 − 22x + 3) is a factor of (1)
Since x2 + 3 − 2√2x is factor of (1), Then
x2 + 3 + 2√2x is also a factor of (1)
⇒ (x2 + 3 − 2√2x) (x2 + 3 + 2√2x) is a factor of (1)
x4 + 6x2 + 9 – 8x2.
= x4 – 2x2 + 9 ...(ii)
From (1) & (2), we have
P4 – 2P2 + 9I = 0
⇒ P4 = 2P2 – 9I
Most Upvoted Answer
Let P be a 4 × 4 matrix with entries from the set of rational nu...
Given P is a 4 × 4 matrix with rational entries.
Let characteristic polynomial of P is
hp(x) = x4 + ax3 + bx2 + cx + d, ...(i)
where a, b, c and d are rational.
Since √2 + i is a root of (1), Then √2 − i is also a root of (1)
⇒ (x2 − 22x + 3) is a factor of (1)
Since x2 + 3 − 2√2x is factor of (1), Then
x2 + 3 + 2√2x is also a factor of (1)
⇒ (x2 + 3 − 2√2x) (x2 + 3 + 2√2x) is a factor of (1)
x4 + 6x2 + 9 – 8x2.
= x4 – 2x2 + 9 ...(ii)
From (1) & (2), we have
P4 – 2P2 + 9I = 0
⇒ P4 = 2P2 – 9I
Let characteristic polynomial of P is
hp(x) = x4 + ax3 + bx2 + cx + d, ...(i)
where a, b, c and d are rational.
Since √2 + i is a root of (1), Then √2 − i is also a root of (1)
⇒ (x2 − 22x + 3) is a factor of (1)
Since x2 + 3 − 2√2x is factor of (1), Then
x2 + 3 + 2√2x is also a factor of (1)
⇒ (x2 + 3 − 2√2x) (x2 + 3 + 2√2x) is a factor of (1)
x4 + 6x2 + 9 – 8x2.
= x4 – 2x2 + 9 ...(ii)
From (1) & (2), we have
P4 – 2P2 + 9I = 0
⇒ P4 = 2P2 – 9I
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Let P be a 4 × 4 matrix with entries from the set of rational numbers. If with is a root of the characteristic polynomial of P and I is the 4 × 4 identity matrix, thena)P4 = 2P2 - 9Ib)P4 = 4P2 - 91 c)P4 = 4P2 + 9Id)P4 = 2P2 + 9ICorrect answer is option 'A'. Can you explain this answer?
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Let P be a 4 × 4 matrix with entries from the set of rational numbers. If with is a root of the characteristic polynomial of P and I is the 4 × 4 identity matrix, thena)P4 = 2P2 - 9Ib)P4 = 4P2 - 91 c)P4 = 4P2 + 9Id)P4 = 2P2 + 9ICorrect 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 P be a 4 × 4 matrix with entries from the set of rational numbers. If with is a root of the characteristic polynomial of P and I is the 4 × 4 identity matrix, thena)P4 = 2P2 - 9Ib)P4 = 4P2 - 91 c)P4 = 4P2 + 9Id)P4 = 2P2 + 9ICorrect 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 P be a 4 × 4 matrix with entries from the set of rational numbers. If with is a root of the characteristic polynomial of P and I is the 4 × 4 identity matrix, thena)P4 = 2P2 - 9Ib)P4 = 4P2 - 91 c)P4 = 4P2 + 9Id)P4 = 2P2 + 9ICorrect answer is option 'A'. Can you explain this answer?.
Let P be a 4 × 4 matrix with entries from the set of rational numbers. If with is a root of the characteristic polynomial of P and I is the 4 × 4 identity matrix, thena)P4 = 2P2 - 9Ib)P4 = 4P2 - 91 c)P4 = 4P2 + 9Id)P4 = 2P2 + 9ICorrect 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 P be a 4 × 4 matrix with entries from the set of rational numbers. If with is a root of the characteristic polynomial of P and I is the 4 × 4 identity matrix, thena)P4 = 2P2 - 9Ib)P4 = 4P2 - 91 c)P4 = 4P2 + 9Id)P4 = 2P2 + 9ICorrect 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 P be a 4 × 4 matrix with entries from the set of rational numbers. If with is a root of the characteristic polynomial of P and I is the 4 × 4 identity matrix, thena)P4 = 2P2 - 9Ib)P4 = 4P2 - 91 c)P4 = 4P2 + 9Id)P4 = 2P2 + 9ICorrect answer is option 'A'. Can you explain this answer?.
Solutions for Let P be a 4 × 4 matrix with entries from the set of rational numbers. If with is a root of the characteristic polynomial of P and I is the 4 × 4 identity matrix, thena)P4 = 2P2 - 9Ib)P4 = 4P2 - 91 c)P4 = 4P2 + 9Id)P4 = 2P2 + 9ICorrect answer is option 'A'. Can you explain this answer? in English & in Hindi are available as part of our courses for IIT JAM.
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Let P be a 4 × 4 matrix with entries from the set of rational numbers. If with is a root of the characteristic polynomial of P and I is the 4 × 4 identity matrix, thena)P4 = 2P2 - 9Ib)P4 = 4P2 - 91 c)P4 = 4P2 + 9Id)P4 = 2P2 + 9ICorrect answer is option 'A'. Can you explain this answer?, a detailed solution for Let P be a 4 × 4 matrix with entries from the set of rational numbers. If with is a root of the characteristic polynomial of P and I is the 4 × 4 identity matrix, thena)P4 = 2P2 - 9Ib)P4 = 4P2 - 91 c)P4 = 4P2 + 9Id)P4 = 2P2 + 9ICorrect answer is option 'A'. Can you explain this answer? has been provided alongside types of Let P be a 4 × 4 matrix with entries from the set of rational numbers. If with is a root of the characteristic polynomial of P and I is the 4 × 4 identity matrix, thena)P4 = 2P2 - 9Ib)P4 = 4P2 - 91 c)P4 = 4P2 + 9Id)P4 = 2P2 + 9ICorrect answer is option 'A'. Can you explain this answer? theory, EduRev gives you an
ample number of questions to practice Let P be a 4 × 4 matrix with entries from the set of rational numbers. If with is a root of the characteristic polynomial of P and I is the 4 × 4 identity matrix, thena)P4 = 2P2 - 9Ib)P4 = 4P2 - 91 c)P4 = 4P2 + 9Id)P4 = 2P2 + 9ICorrect answer is option 'A'. Can you explain this answer? tests, examples and also practice IIT JAM tests.
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