Let the determinant of a square matrix $$A$$ of order $$m$$ be $$m - n$$, where $$m$$ and $$n$$ satisfy $$4m + n = 22$$ and $$17m + 4n = 93$$. If $$det(n \ adj(adj(mA))) = 3^a 5^b 6^c$$, then $$a + b + c$$ is equal to

$$4m + n = 22$$

$$17m + 4n = 93$$

$$m = 5, \quad n = 2$$

$$|A| = m - n = 5 - 2 = 3$$

$$|n \text{ adj}(\text{adj}(mA))| = n^m |\text{adj}(\text{adj}(mA))|$$

$$|n \text{ adj}(\text{adj}(mA))| = n^5 |mA|^{(5-1)^2}$$

$$|n \text{ adj}(\text{adj}(mA))| = 2^5 (m^5 |A|)^{16}$$

$$|n \text{ adj}(\text{adj}(mA))| = 2^5 (5^5 \cdot 3)^{16}$$

$$|n \text{ adj}(\text{adj}(mA))| = 2^5 \cdot 5^{80} \cdot 3^{16}$$

$$|n \text{ adj}(\text{adj}(mA))| = (2^5 \cdot 3^5) \cdot 3^{11} \cdot 5^{80}$$

$$|n \text{ adj}(\text{adj}(mA))| = 3^{11} \cdot 5^{80} \cdot 6^5$$

$$a = 11, \quad b = 80, \quad c = 5$$

$$a + b + c = 11 + 80 + 5 = 96$$