Matrices and Determinants!

Note that $f(t) = \big|A + tB\big|$ is a polynomial of degree $7$ in $t$ , so that $f(t) \; = \; \sum_{j=0}^7 c_jt^j$ where $c_0 = f(0) = |A|$ and $c_7 = |B|$ . Then $\sum_{k=1}^7 f(\omega^{k-1}) \; = \; \sum_{j=0}^7 c_j\sum_{k=1}^7 \omega^{j(k-1)} \; = \; 7(c_0 + c_7) \; = \; 7(|A|+|B|)$ since $\omega^{j}$ is a primitive $7$ th root of unity, so that $\sum_{k=1}^7 \omega^{j(k-1)} = 0$ , for all $1 \le j \le 6$ .

Similarly $\sum_{k=1}^7 g(\omega^{k-1}) \; = \; 7(|A| + |B|)$ where $g(t) = \big|At + B\big|$ , which makes the answer $7 + 7 = \boxed{14}$ . Of course we must assume that $|A|+|B| \neq 0$ .