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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?
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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.
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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?
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