Basic Maths - Grand Test 1 -Q5

+2 votes

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}}

asked Jun 21 in Basic Maths by gbeditor (4,380 points)
reshown Jun 23 by gbeditor

2 Answers

+1 vote
 
Best answer

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 gbeditor (4,380 points)
0 votes
The matrix is symmetric when sum of all row and col is equal in 2*2 matrix .

If matrix is symmetric than eign vector are orthogonal to each other here 1) and 2) dot product zero

So d is correct one.
answered Jun 23 by tsnikhilsharmagate2018 (17,270 points)
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