Question

The generalized Eigen value problem [K- λM]X=0 has a non-zero solution for X. What is the value of λ if K=\(\begin{pmatrix}1&1&1\\1&1&1\\1&1&1\end{pmatrix}\), M=\(\begin{pmatrix}4&4&4\\4&4&4\\4&4&4\end{pmatrix}\)?

(a) 1

(b) \(\frac{1}{2}\)

(c) 4

(d) \(\frac{1}{4}\)

I got this question in an interview for internship.

My doubt stems from Eigen Value and Time Dependent Problems in chapter Single Variable Problems of Finite Element Method

Answer 1

The correct option is (d) \(\frac{1}{4}\)

The best explanation: The generalized Eigen value problem [K- λM]X=0 has a non-zero solution for X if the determinant of the matrix [K- λM] equals zero or K=λM

\(\begin{pmatrix}1&1&1\\1&1&1\\1&1&1\end{pmatrix}\)=λ\(\begin{pmatrix}4&4&4\\4&4&4\\4&4&4\end{pmatrix}\)

Equating corresponding elements, we get 1=4*λ

λ=\(\frac{1}{4}\).