IMC 2007 Problem 3

Call a polynomial $P(x_{1}, \ldots, x_{k})$ good if there exist $2\times 2$ real matrices $A_{1}, \ldots, A_{k}$ such that $P(x_{1}, \ldots, x_{k}) = \det \left(\sum_{i=1}^{k}x_{i}A_{i}\right).$

Find all values of $k$ for which all homogeneous polynomials with $k$ variables of degree 2 are good. (A polynomial is homogeneous if each term has the same total degree.)