# Basic Maths - Grand Test 1 -Q5

Let $P$ be a $2\times 2$ 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

$(1)\begin{bmatrix} 1\\ 0 \end{bmatrix}$                 $(2)\begin{bmatrix} 0\\ 1 \end{bmatrix}$               $(3)\begin{bmatrix} 1\\ 1 \end{bmatrix}$

$(A). {\text{(1) only}}$

$(B). {\text{(2) only}}$

$(C). {\text{(3) only}}$

$(D). {\text{(1) and (2) only}}$

reshown Jun 23

+1 vote

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 24 by (4,380 points)