Question

In matrix algebra, what is the value of a-b if the eigenvector of \(\begin{pmatrix}1&1&1 \\ 1&1&1 \\ 1&1&1 \end{pmatrix}\)  corresponding to eigenvalue three is \(\begin{pmatrix}a \\ b \\ a \end{pmatrix}\)?

(a) 0

(b) 1

(c) 2

(d) 3

I have been asked this question in unit test.

The doubt is from Eigen Value and Time Dependent Problems in section Single Variable Problems of Finite Element Method

Answer 1

Right answer is (a) 0

The best explanation: If X is an eigenvector corresponding to an eigenvalue L of a matrix K, then KX=LX. The eigenvector of \(\begin{pmatrix}1&1&1 \\ 1&1&1 \\ 1&1&1 \end{pmatrix}\) corresponding to eigenvalue three is \(\begin{pmatrix}1 \\ 1 \\ 1 \end{pmatrix}\). Equating the corresponding elements of \(\begin{pmatrix}a \\ b \\ a \end{pmatrix}\) and \(\begin{pmatrix}1 \\ 1 \\ 1 \end{pmatrix}\)

a=b=1

a-b=0.