Question

For which value of k does the matrix A=[4 k; 2 −2] have one real eigenvalue of algebraic multiplicity 2?

let lambda = x

x^2 -2x + (-8-2k) = 0

Answer 1

The eigenvalues of A are x, where det(A-x*Identity) = 0.

$$
\Large{\det \begin{bmatrix}
4-x & k \\2& -2-x
\end{bmatrix}=0
\\ ~\\\iff \\~\\
x^2-2x-2k-8=0

}
$$

Answer 2

now use the quadratic formula

Answer 3

thank you perl :)