Mumbai University > Mechanical Engineering > Sem 4 > Applied Mathematics IV

Marks : 5 M

Year : Dec 2015

The characteristic equation is $ \left[ \begin{array}{ccc} 2-\lambda & 0 & -1 \\ 0 & 2-\lambda & 0 \\ -1 & 0 & 2-\lambda \end{array}\right] \;=\; 0 $

∴$ (2- λ)[(2-λ)^2-0]-0 [0-0]-1[0—1)(2-λ)=0$

∴$ 8 - 12 λ+6λ^2- λ^3+ λ-2=0$

∴$ - λ^3+6λ^2-11λ+6=0$

∴$ λ^3-6λ^2+11λ-6=0$------(1)

Let λ=1 on L.H.S of equation (1)

= 1-6+11-6 = 0

∴ (λ-1) is one of the factor.

Now by synthetic division ,

$ \begin{array}{cc|ccc} 1 & & 1 & -6 & 11 & -6 \\ & & & 1 & -5 & 6 \end{array} \\ \overline{ \hspace{10 cm}} \\ \begin{array}{cc|ccc} \hspace{0.3 cm} & & 1 & -5 & 6 & 0 \end{array} $

∴ $(λ-1) (λ^2-5 λ+6)=0$

∴(λ-1)(λ-2) (λ-3) = 0

∴ λ=1,2,3

∴ Eigen values are 1 ,2, 3