Let $$\begin{bmatrix} 2 & 1 & 0 \\ 1 & 2 & -1 \\ 0 & -1 & 2 \end{bmatrix}$$. If $$|adj(adj(adj(2A)))| = (16)^n$$, then $$n$$ is equal to

To solve this quickly, use the properties of determinants and adjoints for an $$n \times n$$ matrix (here, $$m = 3$$):

Expand the determinant of $$A$$:

$$|A| = 2(4 - 1) - 1(2 - 0) + 0 = 2(3) - 2 = 4$$

For a $$3 \times 3$$ matrix, $$|kA| = k^3|A|$$:

$$|2A| = 2^3 \cdot 4 = 8 \cdot 4 = 32 = 2^5$$

The formula for $$|\text{adj}(\text{adj}(\dots(\text{adj}(M))\dots))|$$ with $$k$$ adjoints is $$|M|^{(m-1)^k}$$.

Here, $$k = 3$$ (three adjoints) and $$m = 3$$:

$$| \text{adj}(\text{adj}(\text{adj}(2A))) | = |2A|^{(3-1)^3} = |2A|^{2^3} = |2A|^8$$

Substitute the value of $$|2A|$$:

$$(2^5)^8 = 2^{40}$$

The problem states this equals $$(16)^n$$:

$$(2^4)^n = 2^{4n}$$

Equating powers:

$$4n = 40 \implies n = 10$$