Question

for what value(s) of k does the matrix A=[0,1;k,1] has -1 and 2 eigenvalues ?

Answer 1

\[A=\left[\begin{matrix}0 & 1\\k&1\end{matrix}\right]\]Do you know how to find the eigenvalues for a matrix?

Answer 2

That's exactly the equation you want. You have a polynomial in \(\lambda\) given by \(\lambda^2-\lambda-k=0\) and you want it to have roots of \(-1\) and \(2\).

Let's construct a polynomial with roots \(-1\) and \(2\). We find that \[(x+1)(x-2)=x^2-x-2\]This looks remarkably similar to the polynomial we have with \(\lambda\). So why not let \(k=2\)? This would give us a polynomial with exactly the roots that we want.

Answer 3

you're welcome.