# Basis maths-linear algebra-Q20

+1 vote

Let be a matrix for which there is a constant k such that the sum of entries in each row and each column is k. Which of the following must be an eigenvector of A

reshown May 16

If K be the sum of each row and column then we get $\begin{bmatrix} 1 \\ 1 \end{bmatrix}$  as a eigen vector with respect to eigen value k.

For example, let

$A_{3\times 3} = \begin{bmatrix} 1 & 2 &3 \\ 2 & 3 & 1\\ 3 & 1 & 2 \end{bmatrix}$

Here, sum of each row & column is 6 ,one of the eigen values of matric A is 6

since the characteristic equation of A is $\lambda^3-6\lambda^2-3\lambda + 18 = 0$

$\begin{bmatrix} 1 &2 &3 \\ 2&3 &1 \\ 3& 1 &2 \end{bmatrix} \begin{bmatrix} 1 \\ 1 \\ 1 \end{bmatrix}= \begin{bmatrix} 6 \\ 6 \\ 6 \end{bmatrix}$

$\begin{bmatrix} 1 &2 &3 \\ 2&3 &1 \\ 3& 1 &2 \end{bmatrix} \begin{bmatrix} 1 \\ 1 \\ 1 \end{bmatrix}= 6\begin{bmatrix} 1 \\ 1 \\ 1 \end{bmatrix}$

option C

answered Jun 11 by (112,760 points)