GATE Civil Engineering (CE) 2018 Shift 2 Solved Paper

Question : 53 of 65

Marks: +1, -0

The matrix

(\table 2 , -4 ; 4 , -2)

has

Solution:

A=[\table 2 , -4 ; 4 , -2]

|A-λ|=0

[\table 2-λ , -4 ; 4 , -2-λ]=0

-4-2 λ+2 λ+λ^{2}+16=0

λ^{2}+12=0

λ=± 2 √{3} i

(complex eigen values)
(1)

λ=2 √{3} i

Consider

(A-λ I) X=0

[\table 2-2 √{3} i , -4 ; 4 , -2-2 √{3} i][\table x_{1} ; x_{2}]=[\table 0 ; 0]

2-2 √{3} i x_{1}=4 x_{2}

\;{x_{1}}/{4}=\;{x_{2}}/{2-2 √{3} i}

[\table x_{1} ; x_{2}]=[\table 4 ; 2-2 √{3} i]

(2)

λ=-2 √{3} i

Consider

(A-λ I) X=0

[\table 2+2 √{3} i , -4 ; 4 , -2+2 √{3} i][\table x_{1} ; x_{2}]=[\table 0 ; 0]

2+2 √{3} i x_{1}=4 x_{2}

\;{x_{1}}/{4}=\;{x_ {2}}/{2+2 √{3} i}

[\table x_{1} ; x_{2}]=[\table 4 ; 2+2 √{3} i]

∴

Complex Eigen values and complex Eigen vectors.

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