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The eigenvalues of the matrix representing the following pair of linear equations
x+iy=0
ix+y = 0
are
are
- a)1+i, 1+i
- b)1− i, 1− i
- c)1, i
- d)1+ i, 1− i
Correct answer is option 'D'. Can you explain this answer?
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The eigenvalues of the matrix representing the following pair of linea...
Option D us definitely correct ..it's tricky question ..both diagonal elements are same so eigen values are 1+-i
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The eigenvalues of the matrix representing the following pair of linea...
Given equations:
The given pair of linear equations are:
1) xi + y = 0
2) ix + y = 0
Matrix representation:
To find the eigenvalues of the matrix representing the given equations, we can write the system of equations in matrix form as:
Ax = 0
where A is the coefficient matrix and x is the vector of variables.
The coefficient matrix A is given by:
A = |i 1|
|1 i|
The vector of variables x is given by:
x = |x|
|y|
Finding eigenvalues:
To find the eigenvalues, we need to solve the equation:
det(A - λI) = 0
where λ is the eigenvalue and I is the identity matrix of the same size as A.
The identity matrix I for a 2x2 matrix is:
I = |1 0|
|0 1|
Substituting the values of A and I into the equation, we get:
| i - λ 1 |
| 1 i - λ |
Taking the determinant of this matrix, we have:
(i - λ)(i - λ) - 1 * 1 = 0
Expanding and simplifying, we get:
(i - λ)^2 - 1 = 0
Solving this quadratic equation, we have:
(i - λ)^2 = 1
Taking the square root on both sides, we get:
i - λ = ±√1
Simplifying further, we have two possible cases:
1) i - λ = 1
2) i - λ = -1
Solving each case separately:
1) i - λ = 1
λ = i - 1
2) i - λ = -1
λ = i + 1
Eigenvalues:
The eigenvalues of the matrix A are:
λ₁ = i - 1
λ₂ = i + 1
Conclusion:
The eigenvalues of the matrix representing the given pair of linear equations are 1 - i and 1 + i. Therefore, the correct answer is option D) 1 - i, 1 + i.
The given pair of linear equations are:
1) xi + y = 0
2) ix + y = 0
Matrix representation:
To find the eigenvalues of the matrix representing the given equations, we can write the system of equations in matrix form as:
Ax = 0
where A is the coefficient matrix and x is the vector of variables.
The coefficient matrix A is given by:
A = |i 1|
|1 i|
The vector of variables x is given by:
x = |x|
|y|
Finding eigenvalues:
To find the eigenvalues, we need to solve the equation:
det(A - λI) = 0
where λ is the eigenvalue and I is the identity matrix of the same size as A.
The identity matrix I for a 2x2 matrix is:
I = |1 0|
|0 1|
Substituting the values of A and I into the equation, we get:
| i - λ 1 |
| 1 i - λ |
Taking the determinant of this matrix, we have:
(i - λ)(i - λ) - 1 * 1 = 0
Expanding and simplifying, we get:
(i - λ)^2 - 1 = 0
Solving this quadratic equation, we have:
(i - λ)^2 = 1
Taking the square root on both sides, we get:
i - λ = ±√1
Simplifying further, we have two possible cases:
1) i - λ = 1
2) i - λ = -1
Solving each case separately:
1) i - λ = 1
λ = i - 1
2) i - λ = -1
λ = i + 1
Eigenvalues:
The eigenvalues of the matrix A are:
λ₁ = i - 1
λ₂ = i + 1
Conclusion:
The eigenvalues of the matrix representing the given pair of linear equations are 1 - i and 1 + i. Therefore, the correct answer is option D) 1 - i, 1 + i.
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The eigenvalues of the matrix representing the following pair of linear equationsx+iy=0ix+y = 0area)1+i, 1+ib)1− i, 1− ic)1, id)1+ i, 1− iCorrect answer is option 'D'. Can you explain this answer?
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