GATE Exam > GATE Questions > Consider a 2 x 2 matrix A, whose characterist...
Start Learning for Free
Consider a 2 x 2 matrix A, whose characteristic equation is given by 2A2 - 5 A + 2I = 0 . Then, the value of 2(Tr A)-3(det A) = ___________
Correct answer is '2'. Can you explain this answer?
| FREE This question is part of | Download PDF Attempt this Test |
Verified Answer
Consider a 2 x 2 matrix A, whose characteristic equation is given by 2...
The characteristic equation o f A is 
According to Cayley - Hamilton theorem, eigenvalues of the matrix will also satisfy the characteristic equation i.e.
According to Cayley - Hamilton theorem, eigenvalues of the matrix will also satisfy the characteristic equation i.e.
Most Upvoted Answer
Consider a 2 x 2 matrix A, whose characteristic equation is given by 2...
Solution:
Given:
- A is a 2 x 2 matrix
- The characteristic equation of A is given by 2A^2 - 5A + 2I = 0
- We need to find the value of 2(Tr A) - 3(det A)
Characteristic Equation:
The characteristic equation of a matrix A is given by:
|A - λI| = 0, where λ is an eigenvalue of A and I is the identity matrix.
Eigenvalues and Eigenvectors:
The eigenvalues of a matrix A are the solutions to the characteristic equation. Let's find the eigenvalues of matrix A.
Given characteristic equation: 2A^2 - 5A + 2I = 0
We can rewrite the equation as:
2(A^2 - (5/2)A) = -2I
(A^2 - (5/2)A) = -I
Taking determinant on both sides:
|A^2 - (5/2)A| = -|I|
|A^2 - (5/2)A| = -1
The determinant of a product of matrices is equal to the product of determinants. So we can write:
|A^2 - (5/2)A| = |A^2 - (5/2)A| = -1
Let λ1 and λ2 be the eigenvalues of matrix A.
So, the characteristic equation can be written as:
(λ - λ1)(λ - λ2) = -1
Expanding the equation, we get:
λ^2 - (λ1 + λ2)λ + λ1λ2 = -1
Comparing the coefficients with the characteristic equation, we have:
λ^2 - (Tr A)λ + det A = 0
Comparing the above equation with the characteristic equation, we get:
Tr A = (5/2)
det A = -1
Calculating the value of 2(Tr A) - 3(det A):
Substituting the values of Tr A and det A in the given expression:
2(Tr A) - 3(det A) = 2(5/2) - 3(-1) = 5 - (-3) = 5 + 3 = 8
However, the correct answer is given as 2.
Explanation:
There seems to be an error in the given answer. The correct value of 2(Tr A) - 3(det A) should be 8, not 2.
Given:
- A is a 2 x 2 matrix
- The characteristic equation of A is given by 2A^2 - 5A + 2I = 0
- We need to find the value of 2(Tr A) - 3(det A)
Characteristic Equation:
The characteristic equation of a matrix A is given by:
|A - λI| = 0, where λ is an eigenvalue of A and I is the identity matrix.
Eigenvalues and Eigenvectors:
The eigenvalues of a matrix A are the solutions to the characteristic equation. Let's find the eigenvalues of matrix A.
Given characteristic equation: 2A^2 - 5A + 2I = 0
We can rewrite the equation as:
2(A^2 - (5/2)A) = -2I
(A^2 - (5/2)A) = -I
Taking determinant on both sides:
|A^2 - (5/2)A| = -|I|
|A^2 - (5/2)A| = -1
The determinant of a product of matrices is equal to the product of determinants. So we can write:
|A^2 - (5/2)A| = |A^2 - (5/2)A| = -1
Let λ1 and λ2 be the eigenvalues of matrix A.
So, the characteristic equation can be written as:
(λ - λ1)(λ - λ2) = -1
Expanding the equation, we get:
λ^2 - (λ1 + λ2)λ + λ1λ2 = -1
Comparing the coefficients with the characteristic equation, we have:
λ^2 - (Tr A)λ + det A = 0
Comparing the above equation with the characteristic equation, we get:
Tr A = (5/2)
det A = -1
Calculating the value of 2(Tr A) - 3(det A):
Substituting the values of Tr A and det A in the given expression:
2(Tr A) - 3(det A) = 2(5/2) - 3(-1) = 5 - (-3) = 5 + 3 = 8
However, the correct answer is given as 2.
Explanation:
There seems to be an error in the given answer. The correct value of 2(Tr A) - 3(det A) should be 8, not 2.
|
Explore Courses for GATE exam
|
|
Consider a 2 x 2 matrix A, whose characteristic equation is given by 2A2 - 5 A + 2I = 0 . Then, the value of 2(Tr A)-3(det A) = ___________Correct answer is '2'. Can you explain this answer?
Question Description
Consider a 2 x 2 matrix A, whose characteristic equation is given by 2A2 - 5 A + 2I = 0 . Then, the value of 2(Tr A)-3(det A) = ___________Correct answer is '2'. Can you explain this answer? for GATE 2024 is part of GATE preparation. The Question and answers have been prepared according to the GATE exam syllabus. Information about Consider a 2 x 2 matrix A, whose characteristic equation is given by 2A2 - 5 A + 2I = 0 . Then, the value of 2(Tr A)-3(det A) = ___________Correct answer is '2'. Can you explain this answer? covers all topics & solutions for GATE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Consider a 2 x 2 matrix A, whose characteristic equation is given by 2A2 - 5 A + 2I = 0 . Then, the value of 2(Tr A)-3(det A) = ___________Correct answer is '2'. Can you explain this answer?.
Consider a 2 x 2 matrix A, whose characteristic equation is given by 2A2 - 5 A + 2I = 0 . Then, the value of 2(Tr A)-3(det A) = ___________Correct answer is '2'. Can you explain this answer? for GATE 2024 is part of GATE preparation. The Question and answers have been prepared according to the GATE exam syllabus. Information about Consider a 2 x 2 matrix A, whose characteristic equation is given by 2A2 - 5 A + 2I = 0 . Then, the value of 2(Tr A)-3(det A) = ___________Correct answer is '2'. Can you explain this answer? covers all topics & solutions for GATE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Consider a 2 x 2 matrix A, whose characteristic equation is given by 2A2 - 5 A + 2I = 0 . Then, the value of 2(Tr A)-3(det A) = ___________Correct answer is '2'. Can you explain this answer?.
Solutions for Consider a 2 x 2 matrix A, whose characteristic equation is given by 2A2 - 5 A + 2I = 0 . Then, the value of 2(Tr A)-3(det A) = ___________Correct answer is '2'. Can you explain this answer? in English & in Hindi are available as part of our courses for GATE.
Download more important topics, notes, lectures and mock test series for GATE Exam by signing up for free.
Here you can find the meaning of Consider a 2 x 2 matrix A, whose characteristic equation is given by 2A2 - 5 A + 2I = 0 . Then, the value of 2(Tr A)-3(det A) = ___________Correct answer is '2'. Can you explain this answer? defined & explained in the simplest way possible. Besides giving the explanation of
Consider a 2 x 2 matrix A, whose characteristic equation is given by 2A2 - 5 A + 2I = 0 . Then, the value of 2(Tr A)-3(det A) = ___________Correct answer is '2'. Can you explain this answer?, a detailed solution for Consider a 2 x 2 matrix A, whose characteristic equation is given by 2A2 - 5 A + 2I = 0 . Then, the value of 2(Tr A)-3(det A) = ___________Correct answer is '2'. Can you explain this answer? has been provided alongside types of Consider a 2 x 2 matrix A, whose characteristic equation is given by 2A2 - 5 A + 2I = 0 . Then, the value of 2(Tr A)-3(det A) = ___________Correct answer is '2'. Can you explain this answer? theory, EduRev gives you an
ample number of questions to practice Consider a 2 x 2 matrix A, whose characteristic equation is given by 2A2 - 5 A + 2I = 0 . Then, the value of 2(Tr A)-3(det A) = ___________Correct answer is '2'. Can you explain this answer? tests, examples and also practice GATE tests.
|
|
Explore Courses for GATE exam
|
|
Signup for Free!
Signup to see your scores go up within 7 days! Learn & Practice with 1000+ FREE Notes, Videos & Tests.
|
© EduRev
|
Education Revolution
|
Follow Us
|
Signup to see your scores
go up within 7 days!
Access 1000+ FREE Docs, Videos and Tests
Takes less than 10 seconds to signup