IMC 2007 · Problem 3

Call a polynomial $P(x_1, \dots, x_k)$ good if there exist $2 \times 2$ real matrices $A_1, \dots, A_k$ such that

$$P(x_1, \dots, x_k) = \det\left( \sum_{i=1}^{k} x_i A_i \right).$$

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.)