Key Concepts and Formulas

1. Determinant of Adjoint Matrix: For any $m \times m$ square matrix $A$, the determinant of its adjoint is given by: $|\operatorname{adj} A| = |A|^{m-1}$ This formula is crucial for relating the determinant of an adjoint matrix back to the determinant of the original matrix.

2. Determinant of Multiple Adjoints: Applying the above property iteratively for a $m \times m$ matrix $A$:

   • $|\operatorname{adj}(\operatorname{adj} A)| = |A|^{(m-1)^2}$
   • $|\operatorname{adj}(\operatorname{adj}(\operatorname{adj} A))| = |A|^{(m-1)^3}$ These extended properties simplify calculations involving multiple adjoint operations.

3. Determinant of a Matrix Power: For any square matrix $A$ and a positive integer $k$: $|A^k| = |A|^k$ This property allows us to move the exponent outside the determinant, simplifying expressions like $\left|A^{\left(n^2+n\right)}\right|$.

Step-by-Step Solution

Step 1: Determine the Value of $|A|$ We begin by finding the determinant of matrix $A$, denoted as $|A|$, using the given condition. Why: The entire problem revolves around $|A|$, so establishing its value is the foundational step.

Given that $A$ is a $3 \times 3$ matrix, its dimension $m=3$. We are provided with the condition $|\operatorname{adj}(\operatorname{adj}(\operatorname{adj} \mathrm{A}))|=81$.

Using the formula for the determinant of triple adjoints: $|\operatorname{adj}(\operatorname{adj}(\operatorname{adj} A))| = |A|^{(m-1)^3}$ Substitute $m=3$: $|\operatorname{adj}(\operatorname{adj}(\operatorname{adj} A))| = |A|^{(3-1)^3} = |A|^{2^3} = |A|^8$ Now, equate this with the given value: $|A|^8 = 81$ To solve for $|A|$, we express $81$ as a power of $3$: $81 = 3^4$. $|A|^8 = 3^4$ Taking the $8^{th}$ root (or raising to the power of $1/8$) on both sides: $(|A|^8)^{1/8} = (3^4)^{1/8}$ $|A| = 3^{4/8} = 3^{1/2} = \sqrt{3}$ Since we are dealing with determinants of real matrices, $|A|$ is a real number. For the expression $|A|^{1/2}$ to be defined as a real value, $|A|$ must be non-negative. Thus, we take the positive root.

Step 2: Solve the Equation to Find the Set $S$ Next, we determine the integer values of $n$ that satisfy the given equation, forming the set $S$. Why: The final calculation requires summing terms for $n \in S$, so identifying these values is crucial.

The given equation is: $(|\operatorname{adj}(\operatorname{adj} A)|)^{\frac{(n-1)^2}{2}}=|A|^{\left(3 n^2-5 n-4\right)}$ First, simplify the left-hand side using the property for the determinant of double adjoints: $|\operatorname{adj}(\operatorname{adj} A)| = |A|^{(m-1)^2}$ Substitute $m=3$: $|\operatorname{adj}(\operatorname{adj} A)| = |A|^{(3-1)^2} = |A|^{2^2} = |A|^4$ Now, substitute this simplified expression back into the given equation: $(|A|^4)^{\frac{(n-1)^2}{2}}=|A|^{\left(3 n^2-5 n-4\right)}$ Using the exponent rule $(x^a)^b = x^{ab}$: $|A|^{4 \cdot \frac{(n-1)^2}{2}}=|A|^{\left(3 n^2-5 n-4\right)}$ $|A|^{2(n-1)^2}=|A|^{\left(3 n^2-5 n-4\right)}$ Since the base $|A|=\sqrt{3}$ is positive and not equal to $1$, we can equate the exponents: $2(n-1)^2 = 3n^2 - 5n - 4$ Expand the left side: $2(n^2 - 2n + 1) = 3n^2 - 5n - 4$ $2n^2 - 4n + 2 = 3n^2 - 5n - 4$ Rearrange the terms to form a quadratic equation by moving all terms to one side: $0 = (3n^2 - 2n^2) + (-5n + 4n) + (-4 - 2)$ $0 = n^2 - n - 6$ Now, factor the quadratic expression to find the values of $n$: $(n-3)(n+2) = 0$ This yields two integer values for $n$: $n-3 = 0 \implies n=3$ $n+2 = 0 \implies n=-2$ The set $S$ consists of these integer values: $S = \{-2, 3\}$

Step 3: Calculate the Final Sum \sum_\limits{n \in S}\left|A^{\left(n^2+n\right)}\right| Finally, we compute the required sum by evaluating the expression $\left|A^{\left(n^2+n\right)}\right|$ for each $n$ in $S$ and adding the results. Why: This is the objective of the problem, bringing together the values of $|A|$ and $S$.

Using the property $|A^k| = |A|^k$, we have $\left|A^{\left(n^2+n\right)}\right| = ||A|^{\left(n^2+n\right)}|$. Since $n^2+n = n(n+1)$ is always an even integer for any integer $n$, $|A|^{\left(n^2+n\right)}$ will always be positive (as $|A|=\sqrt{3}$). Therefore, $||A|^{\left(n^2+n\right)}| = |A|^{\left(n^2+n\right)}$. We use $|A| = \sqrt{3}$.

Let's calculate the term for each $n \in S$:

For $n = -2$: The exponent is $n^2+n = (-2)^2 + (-2) = 4 - 2 = 2$. The term is $|A|^2 = (\sqrt{3})^2 = 3$.

For $n = 3$: The exponent is $n^2+n = (3)^2 + (3) = 9 + 3 = 12$. The term is $|A|^{12} = (\sqrt{3})^{12}$. We can rewrite $(\sqrt{3})^{12}$ as $(3^{1/2})^{12} = 3^{(1/2) \cdot 12} = 3^6$. Calculate $3^6$: $3^1=3$, $3^2=9$, $3^3=27$, $3^4=81$, $3^5=243$, $3^6=729$. So, the term is $729$.

Now, sum these terms: $\sum_{n \in S}\left|A^{\left(n^2+n\right)}\right| = |A|^2 + |A|^{12}$ $= 3 + 729$ $= 732$

Common Mistakes & Tips

• Determinant Properties: Ensure correct application of adjoint determinant formulas, especially for multiple adjoints. A common error is mistakenly using $|A|^{m-1}$ for all levels of adjoints instead of $(m-1)^2$, $(m-1)^3$, etc.
• Algebraic Precision: Be careful with expanding $(n-1)^2$ and solving the quadratic equation. Small sign errors or calculation mistakes can lead to incorrect values for $n$.
• Exponent Rules: When dealing with fractional exponents like $3^{1/2}$ raised to a power, remember $(x^a)^b = x^{ab}$. This is critical for correctly calculating $(\sqrt{3})^{12}$ as $3^6$.

Summary

The problem required us to first determine the determinant of matrix $A$ using its triple adjoint property. We found $|A|=\sqrt{3}$. Then, we solved a given exponential equation involving $n$ by expressing both sides in terms of $|A|$ and equating exponents, leading to a quadratic equation. The integer solutions for $n$ were $3$ and $-2$. Finally, we calculated the sum of $|A|^{n^2+n}$ for these values of $n$, which resulted in $3 + 729 = 732$.

The final answer is $\boxed{\text{732}}$ which corresponds to option (D).