A (left) stochastic matrix is one which has only non-negative entries, and such that the entries in each column sums up to $1$. Let $A$ be any (general) $2 \times 2$ stochastic matrix.

a) Show that one of the eigenvalues of $A$ is $1$.

b) Show that a general $n \times n$ stochastic matrix also has one eigenvalue equal to $1$.

c) What happens if the column sums are $1$ as above, but the entries may be any (possibly negative) real numbers?

I'm not sure how to start with this question. Should I explain using the Perron-Frobenius theorem? How to show for (a)? Because I think I can use the theorem only for (b), or is there a simpler explanation? And I'm not sure how to answer (c) as well.


1 Answer 1


Hint: a matrix $A$ has eigenvalue $1$ if and only if $\det(A-I)=0$. Now look at $A-I$ and see what you get if you evaluate the determinant by adding the rows.


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