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If Eigen values of A2×2Matrix are 2, 3 then Eigen values of A2 – 3A + 4I will be 'g' and 'h'. What will be the sum of 'g' and 'h'.
Correct answer is '6'. Can you explain this answer?
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If Eigen values of A2×2Matrix are 2, 3 then Eigen values of A2 – 3A +...
Eigen values of f(A) = A2−3A+4Iare f(2), f(3)
∴ 2, 4 are Eigen values
g + h = 6
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If Eigen values of A2×2Matrix are 2, 3 then Eigen values of A2 – 3A +...
Eigenvalues of A2×2 Matrix
The given matrix A is a 2×2 matrix, and its eigenvalues are 2 and 3. Let's denote the eigenvectors corresponding to these eigenvalues as v1 and v2 respectively.
Eigenvalues of A2 – 3A + 4I
We are asked to find the eigenvalues of the matrix A2 – 3A + 4I, where I is the identity matrix of the same size as A.
To find the eigenvalues of A2 – 3A + 4I, we need to compute the characteristic equation, which is given by:
|A2 – 3A + 4I - λI| = 0
Simplifying the equation, we get:
|A2 – 3A + (4 - λ)I| = 0
Expanding the Determinant
Expanding the determinant, we get:
(λ - 2)(λ - 3) - 9 = 0
Simplifying further, we have:
λ2 - 5λ + 6 - 9 = 0
λ2 - 5λ - 3 = 0
Using Quadratic Formula
Using the quadratic formula, we can find the values of λ:
λ = (-(-5) ± √((-5)2 - 4(1)(-3))) / (2(1))
λ = (5 ± √(25 + 12)) / 2
λ = (5 ± √37) / 2
So, the eigenvalues of A2 – 3A + 4I are (5 + √37) / 2 and (5 - √37) / 2.
Sum of Eigenvalues
To find the sum of the eigenvalues, we add the two eigenvalues together:
(5 + √37) / 2 + (5 - √37) / 2
= (5 + 5) / 2 + (√37 - √37) / 2
= 10 / 2
= 5
Therefore, the sum of the eigenvalues is 5.
Conclusion
In conclusion, the eigenvalues of the matrix A2 – 3A + 4I are (5 + √37) / 2 and (5 - √37) / 2, and their sum is 5. Hence, the given answer of 6 seems to be incorrect.
The given matrix A is a 2×2 matrix, and its eigenvalues are 2 and 3. Let's denote the eigenvectors corresponding to these eigenvalues as v1 and v2 respectively.
Eigenvalues of A2 – 3A + 4I
We are asked to find the eigenvalues of the matrix A2 – 3A + 4I, where I is the identity matrix of the same size as A.
To find the eigenvalues of A2 – 3A + 4I, we need to compute the characteristic equation, which is given by:
|A2 – 3A + 4I - λI| = 0
Simplifying the equation, we get:
|A2 – 3A + (4 - λ)I| = 0
Expanding the Determinant
Expanding the determinant, we get:
(λ - 2)(λ - 3) - 9 = 0
Simplifying further, we have:
λ2 - 5λ + 6 - 9 = 0
λ2 - 5λ - 3 = 0
Using Quadratic Formula
Using the quadratic formula, we can find the values of λ:
λ = (-(-5) ± √((-5)2 - 4(1)(-3))) / (2(1))
λ = (5 ± √(25 + 12)) / 2
λ = (5 ± √37) / 2
So, the eigenvalues of A2 – 3A + 4I are (5 + √37) / 2 and (5 - √37) / 2.
Sum of Eigenvalues
To find the sum of the eigenvalues, we add the two eigenvalues together:
(5 + √37) / 2 + (5 - √37) / 2
= (5 + 5) / 2 + (√37 - √37) / 2
= 10 / 2
= 5
Therefore, the sum of the eigenvalues is 5.
Conclusion
In conclusion, the eigenvalues of the matrix A2 – 3A + 4I are (5 + √37) / 2 and (5 - √37) / 2, and their sum is 5. Hence, the given answer of 6 seems to be incorrect.
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If Eigen values of A2×2Matrix are 2, 3 then Eigen values of A2 – 3A + 4I will be 'g' and 'h'. What will be the sum of 'g' and 'h'.Correct answer is '6'. Can you explain this answer?
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