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There's this part of my assignment which involves stochastic matrices and i've done most parts of it but there's one part which requires me to show that its eigenvalue is 1. The only way i can think of this is by calculating the det|A-$\lambda$I| and showing that equals to zero when $\lambda$=1. But really how do I calculate a determinant of a 6x6 matrices? Given below is the stochastic matrice that i have found;

\begin{bmatrix} 0 & \frac{1}{2} & \frac{1}{3} & 0 & 0 & 0 \\ \frac{1}{2} & 0 & 0 & \frac{1}{3} & 0 & 0 \\ \frac{1}{2} & 0 & 0 & \frac{1}{3}& \frac{1}{2}& 0 \\ 0 & \frac{1}{2} & \frac{1}{3} & 0 & 0 & \frac{1}{2} \\ 0 & 0 & \frac{1}{3}& 0 & 0 & \frac{1}{2} \\ 0 & 0 & 0 & \frac{1}{3} & \frac{1}{2} & 0 \\ \end{bmatrix}

I don't think i'm suppose to compute this in a really long and complicated way but i also do have to find the eigenvector associated to eigenvalue 1. That's why I'm all confused about this. What would be the way to go about this?

Toby Mak
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1 Answers1

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Subtract $I_6$ from the given matrix $M$, then find the reduced row-echelon form. We get $$\left[\begin{matrix}1 & 0 & 0 & 0 & 0 & -1\\0 & 1 & 0 & 0 & 0 & -1\\0 & 0 & 1 & 0 & 0 & - \frac{3}{2}\\0 & 0 & 0 & 1 & 0 & - \frac{3}{2}\\0 & 0 & 0 & 0 & 1 & -1\\0 & 0 & 0 & 0 & 0 & 0\end{matrix}\right]$$ An eigenvector corresponding to $1$ is a vector in the nullspace of $M-I_6$. The above RREF shows that one such vector is $(1,1,3/2,3/2,1,1)^T$.

Parcly Taxel
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