Well, have you sketched the graph? Perhaps more simply, the characteristic polynomial will be of the form . When . But for sufficiently large positive .

In fact, the same kind of argument shows that A has at least one postive eigenvalue also.

As for the graph, just think of a parabola with y-intercept below the x-axis.

Yes, A and its transpose have the same eigenvalues. What can you say about the corresponding eigenvectors?2. Consider an eigenvalue lambda of an nxn matrix A. we know that lambda is an eignvalue of Atranspose as well (since A and Atranspose have the same characteristic polynomial). Compare the geometric multiplicities of lambda as an eignvalue of A and A transpose

Hint: has eigenvalues 1 and 2. What are its corresponding eigenvectors?

Its transpose, also has eigenvalues 1 and 2. What are its corresponding eigenvectors?