Key Concepts and Formulas

This problem relies on a strong understanding of fundamental determinant properties for a square matrix $A$ of order $m$:

1. Determinant of a Scalar Multiple of a Matrix: If $k$ is a scalar, then $\det(kA) = k^m \det(A)$. This formula highlights how scaling a matrix affects its determinant, with the scalar raised to the power of the matrix's order.
2. Determinant of the Adjoint of a Matrix: The determinant of the adjoint of $A$ is given by $\det(\operatorname{adj}(A)) = (\det(A))^{m-1}$. This property is crucial for handling adjoints in determinant calculations.
3. Determinant of the Nested Adjoint: For a square matrix $A$ of order $m$, the determinant of the adjoint of the adjoint is $\det(\operatorname{adj}(\operatorname{adj}(A))) = (\det(A))^{(m-1)^2}$. This formula is derived by applying the second property twice and is a common shortcut in such problems.

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Step-by-Step Solution

Step 1: Determine the Values of $m$ and $n$

The problem provides a system of two linear equations to find $m$ and $n$:

1. $4m + n = 22$
2. $17m + 4n = 93$

• Action: Our goal is to solve this system. We will use the elimination method. Multiply Equation (1) by 4.

  • Reason: This action makes the coefficient of $n$ in the modified Equation (1) equal to the coefficient of $n$ in Equation (2), allowing us to eliminate $n$ by subtraction. $4 \times (4m + n) = 4 \times 22 \implies 16m + 4n = 88 \quad \text{(Equation 3)}$

• Action: Subtract Equation (3) from Equation (2).

  • Reason: Subtracting the equations will eliminate the $n$ term, leaving a single equation solely in terms of $m$, which can then be easily solved. $(17m + 4n) - (16m + 4n) = 93 - 88$ $17m - 16m = 5$ $m = 5$

• Action: Substitute the value of $m=5$ back into Equation (1).

  • Reason: Knowing $m$, we can now directly calculate the value of $n$ using one of the original equations. $4(5) + n = 22$ $20 + n = 22$ $n = 22 - 20$ $n = 2$

Thus, we have found $m=5$ and $n=2$.

Step 2: Calculate $\det(A)$ and Identify the Order of Matrix $A$

The problem states that $A$ is a square matrix of order $m$.

• Action: Use the value of $m$ determined in Step 1.
  • Reason: The order of the matrix is critical for applying all determinant properties. The order of matrix $A$ is $m=5$.

The determinant of matrix $A$ is given as $\det(A) = m-n$.

• Action: Substitute the values $m=5$ and $n=2$.
  • Reason: This provides the specific numerical value of $\det(A)$ needed for further calculations. $\det(A) = 5 - 2 = 3$

So, we have $\det(A) = 3$ and the order of $A$ is $5$.

Step 3: Evaluate the Determinant Expression $\det(n \operatorname{adj}(\operatorname{adj}(m A)))$

Now we substitute the values $m=5$, $n=2$, and $\det(A)=3$ into the given expression: $\det(n \operatorname{adj}(\operatorname{adj}(m A))) = \det(2 \operatorname{adj}(\operatorname{adj}(5 A)))$

We will evaluate this expression by applying the determinant properties systematically, working from the outermost operation inwards.

• Step 3.1: Apply $\det(kX) = k^{\text{order}} \det(X)$ for the outermost scalar $n=2$.

  • Reason: We treat $X = \operatorname{adj}(\operatorname{adj}(5A))$ as a single matrix. The scalar is $n=2$, and the order of the matrix $X$ is $m=5$. $\det(2 \operatorname{adj}(\operatorname{adj}(5 A))) = 2^m \det(\operatorname{adj}(\operatorname{adj}(5 A))) = 2^5 \det(\operatorname{adj}(\operatorname{adj}(5 A)))$

• Step 3.2: Apply $\det(\operatorname{adj}(\operatorname{adj}(B))) = (\det(B))^{(m-1)^2}$ for the nested adjoints.

  • Reason: Here, the matrix $B$ is $5A$. The order of $B$ is $m=5$. $\det(\operatorname{adj}(\operatorname{adj}(5 A))) = (\det(5 A))^{(m-1)^2} = (\det(5 A))^{(5-1)^2} = (\det(5 A))^{4^2} = (\det(5 A))^{16}$ Substituting this back into our expression from Step 3.1: $2^5 (\det(5 A))^{16}$

• Step 3.3: Apply $\det(kX) = k^{\text{order}} \det(X)$ again for $\det(5A)$.

  • Reason: Here, the scalar is $5$, and the matrix is $A$. The order of $A$ is $m=5$. $\det(5 A) = 5^m \det(A) = 5^5 \det(A)$ Substitute this back into the expression from Step 3.2: $2^5 (5^5 \det(A))^{16}$

• Step 3.4: Substitute the value of $\det(A)$ and simplify using exponent rules.

  • Reason: We now have the expression entirely in terms of known numerical values. We know $\det(A) = 3$. $2^5 (5^5 \cdot 3)^{16}$ Using the exponent rule $(xy)^p = x^p y^p$: $2^5 \cdot (5^5)^{16} \cdot 3^{16}$ Using the exponent rule $(x^p)^q = x^{pq}$: $2^5 \cdot 5^{5 \times 16} \cdot 3^{16}$ $2^5 \cdot 5^{80} \cdot 3^{16}$

Step 4: Express the Result in the Desired Form and Find $a+b+c$

The problem asks us to express the final result in the form $3^a 5^b 6^c$. Our current result is $2^5 \cdot 3^{16} \cdot 5^{80}$.

• Action: Manipulate the prime factors to match the target form.

  • Reason: The target form includes $6^c$. Since $6 = 2 \times 3$, we need to group factors of $2$ and $3$ to create $6$. We have $2^5$ and $3^{16}$. We can take $2^5$ and $3^5$ to form $6^5$. $2^5 \cdot 3^{16} = (2^5 \cdot 3^5) \cdot 3^{16-5} = (2 \cdot 3)^5 \cdot 3^{11} = 6^5 \cdot 3^{11}$ Now, substitute this back into our complete expression: $6^5 \cdot 3^{11} \cdot 5^{80}$

• Action: Rearrange the terms to match the format $3^a 5^b 6^c$. $3^{11} \cdot 5^{80} \cdot 6^5$

• Action: Compare the exponents to find $a, b, c$.

  • Reason: This directly gives us the values requested by the problem. By comparing $3^{11} \cdot 5^{80} \cdot 6^5$ with $3^a 5^b 6^c$: $a = 11$ $b = 80$ $c = 5$

• Action: Calculate $a+b+c$.

  • Reason: This is the final value the problem asks for. $a+b+c = 11 + 80 + 5 = 96$

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Common Mistakes & Tips

• Mismatched Order ($m$): A common error is using an incorrect value for $m$ (the order of the matrix) in the determinant formulas, especially when dealing with scalar multiples or adjoints. Always ensure $m$ is correctly identified and consistently used.
• Exponent Rule Errors: Be very careful with exponent rules, particularly $(x^p)^q = x^{pq}$ and $(xy)^p = x^p y^p$. A small mistake here can lead to a significantly different final answer.
• Incorrect Adjoint Formulas: Ensure you correctly apply $\det(\operatorname{adj}(A)) = (\det(A))^{m-1}$ and its derivative for nested adjoints, $\det(\operatorname{adj}(\operatorname{adj}(A))) = (\det(A))^{(m-1)^2}$.
• Final Form Conversion: Pay close attention to the requested format of the final answer (e.g., $3^a 5^b 6^c$). You might need to manipulate prime factors (like converting $2^x \cdot 3^y$ into $6^z \cdot 3^{y-z}$) to match the desired bases.

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Summary

This problem effectively tests your ability to integrate algebraic skills with advanced determinant properties. The solution involved first solving a system of linear equations to determine the matrix order $m$ and the scalar $n$. Then, these values were used to find the determinant of matrix $A$. Finally, the core of the problem was solved by systematically applying determinant properties for scalar multiplication and nested adjoints, carefully managing exponents, and converting the final numerical expression into the required $3^a 5^b 6^c$ format to find the sum $a+b+c$. The key takeaway is the importance of a structured approach, meticulous application of formulas, and precise algebraic manipulation.

The final answer is $\boxed{\text{96}}$, which corresponds to option (A).