GATE CE 2015 Set 1 — Question 36
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Engineering Mathematics → Linear Algebra → Eigenvalues & Eigenvectors
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Question
The smallest and largest Eigen values of the following matrix are:
342−2−4−3265Correct answer
(D) 1.0 and 2.0
Solution
1.Matrix: A=342−2−4−3265. 2.Characteristic Equation: ∣A−λI∣=0. λ3−(Trace A)λ2+(Sum of principal minors)λ−∣A∣=0.
3.Trace: 3+(−4)+5=4. 4.Sum of principal minors:
M11=−4−365=−20+18=−2.
M22=3225=15−4=11.
M33=34−2−4=−12+8=−4.
Sum
=−2+11−4=5.
5.Determinant: ∣A∣=3(−20+18)−(−2)(20−12)+2(−12+8)=−6+16−8=2. 6.Equation: λ3−4λ2+5λ−2=0. 7.Roots: By inspection, λ=1 is a root (1−4+5−2=0). Dividing by
(λ−1):
λ2−3λ+2=0⇒(λ−1)(λ−2)=0.
Eigenvalues are
1,1,2.
8.Smallest and Largest: Smallest =1.0, Largest =2.0. The correct option is (D).
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