Mechanical Engineering Exam > Mechanical Engineering Questions > The trace and determinate of a 2 ×2 matr...
Start Learning for Free
The trace and determinate of a 2 ×2 matrix are known to be – 2 and – 35 respectively. Its eigenvalues are
- a)-30 and – 5
- b)– 35 and – 1
- c)– 7 and 5
- d)17.5 and - 2
Correct answer is option 'C'. Can you explain this answer?
Verified Answer
The trace and determinate of a 2 ×2 matrix are known to be –...
Most Upvoted Answer
The trace and determinate of a 2 ×2 matrix are known to be –...
Trace and Determinant of a 2x2 Matrix:
The trace and determinant of a 2x2 matrix can be determined using the properties of matrices. Let's denote the matrix as A:
A = [[a, b], [c, d]]
where a, b, c, and d are the elements of the matrix.
The trace of the matrix is the sum of the elements on the main diagonal, which in this case is the sum of a and d:
Trace(A) = a + d
The determinant of the matrix is given by the formula:
Determinant(A) = ad - bc
Given Information:
In this question, we are given that the trace of the matrix is 2 and the determinant is 35. So we have:
Trace(A) = 2
Determinant(A) = 35
Finding the Eigenvalues:
To find the eigenvalues of the matrix, we need to solve the characteristic equation, which is given by:
|A - λI| = 0
where A is the matrix, λ is the eigenvalue, and I is the identity matrix.
For a 2x2 matrix A, the characteristic equation is:
|a-λ b| = (a-λ)(d-λ) - bc = 0
|c d-λ|
Expanding the determinant, we get:
(a-λ)(d-λ) - bc = 0
ad - aλ - dλ + λ^2 - bc = 0
λ^2 - (a+d)λ + ad - bc = 0
Substituting the values from the given information, we have:
λ^2 - 2λ + 35 = 0
Solving the Quadratic Equation:
To solve the quadratic equation, we can factorize it or use the quadratic formula. In this case, factoring is not possible, so we'll use the quadratic formula:
λ = (-b ± √(b^2 - 4ac)) / 2a
where a = 1, b = -2, and c = 35.
Substituting the values, we have:
λ = (-(-2) ± √((-2)^2 - 4(1)(35))) / (2(1))
λ = (2 ± √(4 - 140)) / 2
λ = (2 ± √(-136)) / 2
λ = (2 ± i√136) / 2
λ = 1 ± i√34
Conclusion:
From the quadratic equation, we found that the eigenvalues of the matrix are 1 + i√34 and 1 - i√34. However, the given options for the eigenvalues are 7 and 5. Therefore, none of the given options are correct.
Hence, the correct answer cannot be determined from the given options.
The trace and determinant of a 2x2 matrix can be determined using the properties of matrices. Let's denote the matrix as A:
A = [[a, b], [c, d]]
where a, b, c, and d are the elements of the matrix.
The trace of the matrix is the sum of the elements on the main diagonal, which in this case is the sum of a and d:
Trace(A) = a + d
The determinant of the matrix is given by the formula:
Determinant(A) = ad - bc
Given Information:
In this question, we are given that the trace of the matrix is 2 and the determinant is 35. So we have:
Trace(A) = 2
Determinant(A) = 35
Finding the Eigenvalues:
To find the eigenvalues of the matrix, we need to solve the characteristic equation, which is given by:
|A - λI| = 0
where A is the matrix, λ is the eigenvalue, and I is the identity matrix.
For a 2x2 matrix A, the characteristic equation is:
|a-λ b| = (a-λ)(d-λ) - bc = 0
|c d-λ|
Expanding the determinant, we get:
(a-λ)(d-λ) - bc = 0
ad - aλ - dλ + λ^2 - bc = 0
λ^2 - (a+d)λ + ad - bc = 0
Substituting the values from the given information, we have:
λ^2 - 2λ + 35 = 0
Solving the Quadratic Equation:
To solve the quadratic equation, we can factorize it or use the quadratic formula. In this case, factoring is not possible, so we'll use the quadratic formula:
λ = (-b ± √(b^2 - 4ac)) / 2a
where a = 1, b = -2, and c = 35.
Substituting the values, we have:
λ = (-(-2) ± √((-2)^2 - 4(1)(35))) / (2(1))
λ = (2 ± √(4 - 140)) / 2
λ = (2 ± √(-136)) / 2
λ = (2 ± i√136) / 2
λ = 1 ± i√34
Conclusion:
From the quadratic equation, we found that the eigenvalues of the matrix are 1 + i√34 and 1 - i√34. However, the given options for the eigenvalues are 7 and 5. Therefore, none of the given options are correct.
Hence, the correct answer cannot be determined from the given options.
|
Explore Courses for Mechanical Engineering exam
|
|
Similar Mechanical Engineering Doubts
Top Courses for Mechanical EngineeringView all
Top Courses for Mechanical Engineering
Question Description
The trace and determinate of a 2 ×2 matrix are known to be – 2 and – 35 respectively. Its eigenvalues area)-30 and – 5 b)– 35 and – 1 c)– 7 and 5 d)17.5 and - 2 Correct answer is option 'C'. Can you explain this answer? for Mechanical Engineering 2025 is part of Mechanical Engineering preparation. The Question and answers have been prepared according to the Mechanical Engineering exam syllabus. Information about The trace and determinate of a 2 ×2 matrix are known to be – 2 and – 35 respectively. Its eigenvalues area)-30 and – 5 b)– 35 and – 1 c)– 7 and 5 d)17.5 and - 2 Correct answer is option 'C'. Can you explain this answer? covers all topics & solutions for Mechanical Engineering 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for The trace and determinate of a 2 ×2 matrix are known to be – 2 and – 35 respectively. Its eigenvalues area)-30 and – 5 b)– 35 and – 1 c)– 7 and 5 d)17.5 and - 2 Correct answer is option 'C'. Can you explain this answer?.
The trace and determinate of a 2 ×2 matrix are known to be – 2 and – 35 respectively. Its eigenvalues area)-30 and – 5 b)– 35 and – 1 c)– 7 and 5 d)17.5 and - 2 Correct answer is option 'C'. Can you explain this answer? for Mechanical Engineering 2025 is part of Mechanical Engineering preparation. The Question and answers have been prepared according to the Mechanical Engineering exam syllabus. Information about The trace and determinate of a 2 ×2 matrix are known to be – 2 and – 35 respectively. Its eigenvalues area)-30 and – 5 b)– 35 and – 1 c)– 7 and 5 d)17.5 and - 2 Correct answer is option 'C'. Can you explain this answer? covers all topics & solutions for Mechanical Engineering 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for The trace and determinate of a 2 ×2 matrix are known to be – 2 and – 35 respectively. Its eigenvalues area)-30 and – 5 b)– 35 and – 1 c)– 7 and 5 d)17.5 and - 2 Correct answer is option 'C'. Can you explain this answer?.
Solutions for The trace and determinate of a 2 ×2 matrix are known to be – 2 and – 35 respectively. Its eigenvalues area)-30 and – 5 b)– 35 and – 1 c)– 7 and 5 d)17.5 and - 2 Correct answer is option 'C'. Can you explain this answer? in English & in Hindi are available as part of our courses for Mechanical Engineering.
Download more important topics, notes, lectures and mock test series for Mechanical Engineering Exam by signing up for free.
Here you can find the meaning of The trace and determinate of a 2 ×2 matrix are known to be – 2 and – 35 respectively. Its eigenvalues area)-30 and – 5 b)– 35 and – 1 c)– 7 and 5 d)17.5 and - 2 Correct answer is option 'C'. Can you explain this answer? defined & explained in the simplest way possible. Besides giving the explanation of
The trace and determinate of a 2 ×2 matrix are known to be – 2 and – 35 respectively. Its eigenvalues area)-30 and – 5 b)– 35 and – 1 c)– 7 and 5 d)17.5 and - 2 Correct answer is option 'C'. Can you explain this answer?, a detailed solution for The trace and determinate of a 2 ×2 matrix are known to be – 2 and – 35 respectively. Its eigenvalues area)-30 and – 5 b)– 35 and – 1 c)– 7 and 5 d)17.5 and - 2 Correct answer is option 'C'. Can you explain this answer? has been provided alongside types of The trace and determinate of a 2 ×2 matrix are known to be – 2 and – 35 respectively. Its eigenvalues area)-30 and – 5 b)– 35 and – 1 c)– 7 and 5 d)17.5 and - 2 Correct answer is option 'C'. Can you explain this answer? theory, EduRev gives you an
ample number of questions to practice The trace and determinate of a 2 ×2 matrix are known to be – 2 and – 35 respectively. Its eigenvalues area)-30 and – 5 b)– 35 and – 1 c)– 7 and 5 d)17.5 and - 2 Correct answer is option 'C'. Can you explain this answer? tests, examples and also practice Mechanical Engineering tests.
|
|
Explore Courses for Mechanical Engineering exam
|
|
Signup to solve all Doubts
Signup to see your scores go up within 7 days! Learn & Practice with 1000+ FREE Notes, Videos & Tests.
|
© EduRev
|
Education Revolution
|
|
Signup to see your scores
go up
within 7 days!
within 7 days!
Takes less than 10 seconds to signup