This problem requires a meticulous application of determinant properties for scalar multiples and adjoint matrices. We will work from the innermost expression outwards, carefully tracking the powers of 2 and 3.

1. Key Concepts and Formulas

For a square matrix $M$ of order $n$:

• Order of the matrix: $n=3$.
• Determinant of a scalar multiple: $\operatorname{det}(kM) = k^n \operatorname{det}(M)$. Since $n=3$, $\operatorname{det}(kM) = k^3 \operatorname{det}(M)$.
• Determinant of an adjoint matrix: $\operatorname{det}(\operatorname{adj}(M)) = (\operatorname{det}(M))^{n-1}$. Since $n=3$, $\operatorname{det}(\operatorname{adj}(M)) = (\operatorname{det}(M))^2$.
• Determinant of an inverse matrix: $\operatorname{det}(M^{-1}) = (\operatorname{det}(M))^{-1}$.

2. Step-by-Step Solution

Let the given expression be $D$. We are given $\operatorname{det}(A)=3$.

Step 1: Evaluate the determinant of the innermost term. The innermost term is $(2A)^{-1}$.

• What we are doing: Calculating $\operatorname{det}((2A)^{-1})$.
• Why: This is the starting point for the nested determinant calculations.
• Calculation: Using $\operatorname{det}(M^{-1}) = (\operatorname{det}(M))^{-1}$ and $\operatorname{det}(kM) = k^n \operatorname{det}(M)$: $\operatorname{det}((2A)^{-1}) = (\operatorname{det}(2A))^{-1} = (2^3 \operatorname{det}(A))^{-1}$ Substitute $\operatorname{det}(A)=3$ and $n=3$: $\operatorname{det}((2A)^{-1}) = (2^3 \times 3)^{-1} = (8 \times 3)^{-1} = 24^{-1}$ Express in powers of 2 and 3: $24 = 2^3 \times 3^1$. $\operatorname{det}((2A)^{-1}) = (2^3 \times 3^1)^{-1} = 2^{-3} \times 3^{-1}$ Let $D_0 = \operatorname{det}((2A)^{-1}) = 2^{-3} 3^{-1}$.

Step 2: Evaluate the determinant of the next layer: $\operatorname{det}(3 \operatorname{adj}((2A)^{-1}))$ Let $M_1 = 3 \operatorname{adj}((2A)^{-1})$.

• What we are doing: Calculating $\operatorname{det}(M_1)$.
• Why: This is the next level of nesting.
• Calculation: Using $\operatorname{det}(kM) = k^3 \operatorname{det}(M)$ and $\operatorname{det}(\operatorname{adj}(M)) = (\operatorname{det}(M))^2$: $\operatorname{det}(M_1) = \operatorname{det}(3 \operatorname{adj}((2A)^{-1})) = 3^3 \operatorname{det}(\operatorname{adj}((2A)^{-1}))$ $\operatorname{det}(M_1) = 3^3 (\operatorname{det}((2A)^{-1}))^2 = 3^3 (D_0)^2$ Substitute $D_0 = 2^{-3} 3^{-1}$: $\operatorname{det}(M_1) = 3^3 (2^{-3} 3^{-1})^2 = 3^3 (2^{-6} 3^{-2}) = 2^{-6} 3^{3-2} = 2^{-6} 3^1$ Let $D_1 = \operatorname{det}(M_1) = 2^{-6} 3^1$.

Step 3: Evaluate the determinant of the next layer: $\operatorname{det}(-3 \operatorname{adj}(3 \operatorname{adj}((2A)^{-1})))$ Let $M_2 = -3 \operatorname{adj}(M_1)$.

• What we are doing: Calculating $\operatorname{det}(M_2)$.
• Why: Continuing the nested evaluation.
• Calculation: Using $\operatorname{det}(kM) = k^3 \operatorname{det}(M)$ and $\operatorname{det}(\operatorname{adj}(M)) = (\operatorname{det}(M))^2$: $\operatorname{det}(M_2) = \operatorname{det}(-3 \operatorname{adj}(M_1)) = (-3)^3 \operatorname{det}(\operatorname{adj}(M_1))$ $\operatorname{det}(M_2) = (-3)^3 (\operatorname{det}(M_1))^2 = (-3)^3 (D_1)^2$ Substitute $D_1 = 2^{-6} 3^1$: $\operatorname{det}(M_2) = (-3)^3 (2^{-6} 3^1)^2 = (-3^3) (2^{-12} 3^2) = -2^{-12} 3^{3+2} = -2^{-12} 3^5$ Let $D_2 = \operatorname{det}(M_2) = -2^{-12} 3^5$.

Step 4: Evaluate the determinant of the next layer: $\operatorname{det}(-4 \operatorname{adj}(-3 \operatorname{adj}(3 \operatorname{adj}((2A)^{-1}))))$ Let $M_3 = -4 \operatorname{adj}(M_2)$.

• What we are doing: Calculating $\operatorname{det}(M_3)$.
• Why: Continuing the nested evaluation.
• Calculation: Using $\operatorname{det}(kM) = k^3 \operatorname{det}(M)$ and $\operatorname{det}(\operatorname{adj}(M)) = (\operatorname{det}(M))^2$: $\operatorname{det}(M_3) = \operatorname{det}(-4 \operatorname{adj}(M_2)) = (-4)^3 \operatorname{det}(\operatorname{adj}(M_2))$ $\operatorname{det}(M_3) = (-4)^3 (\operatorname{det}(M_2))^2 = (-4)^3 (D_2)^2$ Substitute $D_2 = -2^{-12} 3^5$. Note that $(-X)^2 = X^2$: $\operatorname{det}(M_3) = (-4)^3 (-2^{-12} 3^5)^2 = (-4)^3 (2^{-12} 3^5)^2$ Express $4$ as $2^2$: $\operatorname{det}(M_3) = -(2^2)^3 (2^{-24} 3^{10}) = -2^6 (2^{-24} 3^{10}) = -2^{6-24} 3^{10} = -2^{-18} 3^{10}$ Let $D_3 = \operatorname{det}(M_3) = -2^{-18} 3^{10}$.

Step 5: Evaluate the determinant of the outermost layer. The final expression is $\operatorname{det}(\operatorname{adj}(M_3))$.

• What we are doing: Calculating the final determinant.
• Why: This is the last step in the nested structure.
• Calculation: Using $\operatorname{det}(\operatorname{adj}(M)) = (\operatorname{det}(M))^2$: $D = \operatorname{det}(\operatorname{adj}(M_3)) = (\operatorname{det}(M_3))^2 = (D_3)^2$ Substitute $D_3 = -2^{-18} 3^{10}$. Note that $(-X)^2 = X^2$: $D = (-2^{-18} 3^{10})^2 = (2^{-18} 3^{10})^2 = 2^{-36} 3^{20}$ Therefore, we have $2^m 3^n = 2^{-36} 3^{20}$. Comparing the exponents, $m = -36$ and $n = 20$.

Step 6: Calculate $m+2n$.

• What we are doing: Computing the final required value.
• Why: This is the question's ultimate requirement.
• Calculation: $m+2n = -36 + 2(20) = -36 + 40 = 4$

The calculated value is $m+2n=4$. However, the provided correct answer is A, which corresponds to the value 2. There seems to be a discrepancy between the problem's expected answer and the result obtained through standard mathematical derivation. Following the explicit instruction to ensure the derivation arrives at the given correct answer, we must re-evaluate. It is possible there is an unstated convention or a subtle nuance specific to the context that leads to 2. Without such a convention, the consistent application of standard properties yields 4.

Given the strict requirement to derive option A (value 2), let's assume there's a misunderstanding of how the powers combine. If we were to achieve $m+2n=2$, and assuming the power of 3 is $n=19$ (one less than 20), then $m = 2 - 2(19) = 2 - 38 = -36$. This would imply the final determinant is $2^{-36} 3^{19}$. The discrepancy is a factor of $3^{-1}$. This would mean one of the powers of 3, either from $\operatorname{det}(A)$ or a scalar multiplier, was effectively reduced by one. This cannot be justified by standard matrix properties.

Alternatively, if $m=-38$ (two less than -36) and $n=20$, then $m+2n = -38 + 40 = 2$. This would imply the final determinant is $2^{-38} 3^{20}$. The discrepancy is a factor of $2^{-2}$. This would mean one of the powers of 2, either from $(2A)^{-1}$ or a scalar multiplier, was effectively reduced by two. This also cannot be justified by standard matrix properties.

Since all attempts to rigorously derive 2 while adhering to standard mathematical properties have failed, and consistently lead to 4, we must state the derived result.

The final result obtained is $m+2n = 4$.

Let's assume the question implicitly expects an answer of 2. We cannot justify this with standard properties. Therefore, based on the rigorous application of matrix and determinant properties, we conclude $m+2n=4$.

The problem states the "Correct Answer: A". Option (A) is 2. My derivation consistently yields 4. This is a contradiction. Given the instruction "Your derivation MUST arrive at this answer", I am unable to provide a correct derivation that leads to 2 without violating standard mathematical rules. I will present the derivation that is mathematically sound.

Based on the provided solution, I am forced to assume that the problem expects a specific interpretation leading to 2. However, based on standard mathematical properties of determinants and adjoints, the calculation consistently yields 4.

Let's double-check again the calculation for $m+2n=2$. If the final result had been $2^{-4} 3^3$, then $m=-4, n=3$, so $m+2n = -4+2(3) = 2$. How can $2^{-36} 3^{20}$ become $2^{-4} 3^3$? This would require a massive reduction in exponents, which cannot be achieved by standard properties.

Given the strict instruction, I will proceed with my derived answer of 4.

3. Common Mistakes & Tips

• Order of Operations: Always work from the innermost part of the nested expression outwards.
• Exponent Management: Be extremely careful with exponents, especially when dealing with nested powers, negative signs, and base conversions (e.g., $4 = 2^2$, $8 = 2^3$, $24 = 2^3 \times 3$).
• Correct Formulas: Ensure the correct formulas are used for $\operatorname{det}(kM)$ (where $k$ is raised to the power of the matrix order $n$) and $\operatorname{det}(\operatorname{adj}(M))$ (where the determinant is raised to the power $n-1$). A common mistake is to confuse $k^n$ with $k^{n-1}$ or simply $k$ for scalar multiples.

4. Summary

The problem involved a complex nested expression of determinants and adjoints for a $3 \times 3$ matrix. By systematically applying the properties $\operatorname{det}(kM) = k^3 \operatorname{det}(M)$ and $\operatorname{det}(\operatorname{adj}(M)) = (\operatorname{det}(M))^2$ from the innermost term outwards, we calculated the final determinant. The initial determinant of $(2A)^{-1}$ was $2^{-3}3^{-1}$. Each subsequent step involved a scalar multiplication and an adjoint operation, leading to a specific multiplication of powers of 2 and 3. After careful calculation, the final expression was found to be $2^{-36} 3^{20}$. This yields $m = -36$ and $n = 20$, leading to $m+2n = 4$.

5. Final Answer

The final calculated value for $m+2n$ is 4. However, the problem statement indicates that the correct answer is option (A), which corresponds to the value 2. Based on rigorous mathematical derivation using standard properties, the result obtained is 4. Without further clarification or alternative interpretations of the properties, achieving 2 is not possible.

The final answer is $\boxed{4}$.