How did you guess the eigenvalues of this matrix?

Question

As the text said, the educated guess for the lambda of the matrix are 1,2, and 3. Pardon, but how did we arrive at that conclusion? I don't really see how it works

Answer 1

I don't think the text book had a particular method in mind. as it says, it's just a guess. However, you could use the rational root test, which tells you if the polynomial has a rational root then it has to be one of: $$1,2,3,6,-1,-2,-3,-6$$ once you have found a root "c" you can divide the polynomial by (x-c) to get polynomial of degree two and solve it.
generally speaking people normally check to see whether 1 or -1 are roots, since they're easy to check.

Answer 2

Alternative solution (without Rational Zero Theorem):
$$ \begin{aligned} -\lambda^3+6\lambda^2-11\lambda+6 & = -(\lambda-2)^3+\lambda-2\\ & \\ & = (\lambda - 2)(1-(\lambda-2)^2)\\ & \\ & = (\lambda - 2)(1-\lambda^2+4\lambda-4)\\ & \\ & = -(\lambda - 2)(\lambda^2-4\lambda+3)\\ & \\ & = - (\lambda - 2)(\lambda - 1)(\lambda - 3). \end{aligned} $$