Find matrix given eigen values and two rows

Question

I need to find a matrix $A$ such that $A = \begin{bmatrix} 0 & 1 & 0 \\ 0 & 0 & 1 \\ k_1 & k_2 & k_3 \end{bmatrix}$ such that all eigen values have real part negative and all entries of the matrix are real.

I have no idea on how to do this, and would really appreciate some direction.

Answer 1

Hint:

Take for example $\;f(x)=(x+1)(x+2)(x+3)\;$ , and then $\;k_1,\,k_2,\,k_3\;$ are minus the free coefficient, the linear one and the quadratic one, respectively, of $\;f\;$ .

You could also google "companion matrix", which many times is defined as the transpose of your matrix...but the determinant is the same, of course.