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If the entries in each column of a square matrix M add up to 1, then an eigen value of M is

A) 4  B) 3  C) 2  D) 1

2 Answers

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Consider the 2×2 matrix A=$\begin{bmatrix} a &b \\ c& d \end{bmatrix}$

$\lambda^2 -trace(A)  \lambda+det(A)=0$

$\lambda^2 -(a+d  )\lambda+(ad-bc)=0$

Put $\lambda$=1

$1-a-d+ad-(1-d)(1-a)=0$

$\text{as we know that a+c=1 and b+d=1}$

$1-a-d+ad-1+d+a-ad =0$

$0=0$ other option $\lambda=2,3,4$doesn't satisfy

$\mathbf{(D )ans}$

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