Problem: Let A be a matrix of order $3 \times 3$ and $\det (A) = 2$. Then $\det (\det (A) \text{adj} (5 \text{adj} (A^3)))$ is equal to _____________.

Given Information:

• Matrix A is of order $3 \times 3$, so the dimension $n=3$.
• The determinant of A is $\det(A) = 2$.

Objective: We need to evaluate the expression $\det (\det (A) \text{adj} (5 \text{adj} (A^3)))$.

Substituting the given value of $\det(A)=2$ into the expression, our target is to calculate: $\det (2 \text{adj} (5 \text{adj} (A^3)))$ This problem requires a strong understanding of properties of determinants, especially those involving adjoint matrices and scalar multiplication. We will systematically break down the expression using these properties, working from the innermost part outwards.

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Key Concepts and Formulas

For an $n \times n$ matrix $M$ and a scalar $k$:

1. Determinant of a scalar multiple: $\det(kM) = k^n \det(M)$

   • Explanation: When every element of an $n \times n$ matrix $M$ is multiplied by a scalar $k$, it's equivalent to multiplying each of the $n$ rows (or columns) by $k$. When taking the determinant, $k$ can be factored out from each of these $n$ rows/columns, resulting in $k^n$.

2. Determinant of the adjoint matrix: $\det(\text{adj}(M)) = (\det(M))^{n-1}$

   • Explanation: This fundamental property directly relates the determinant of the adjoint of a matrix to the determinant of the matrix itself. It's a crucial formula for problems involving adjoints.

3. Determinant of a matrix power: $\det(M^k) = (\det(M))^k$

   • Explanation: The determinant of a product of matrices is the product of their determinants. Therefore, $\det(M^k) = \det(M \cdot M \cdot \ldots \cdot M \text{ (k times)}) = \det(M) \cdot \det(M) \cdot \ldots \cdot \det(M) \text{ (k times)} = (\det(M))^k$.

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Step-by-Step Evaluation

We will evaluate the expression $\det (2 \text{adj} (5 \text{adj} (A^3)))$ by simplifying it layer by layer, starting from the innermost term. Remember that $n=3$ for matrix A.

Step 1: Evaluate $\det(A^3)$

• Concept Used: Determinant of a matrix power, $\det(M^k) = (\det(M))^k$.
• Why: This allows us to find the determinant of $A^3$ directly using the given $\det(A)$.
• Calculation: Given $\det(A) = 2$ and the power $k=3$. $\det(A^3) = (\det(A))^3 = 2^3 = 8$ So, $\det(A^3) = 8$.

Step 2: Evaluate $\det(\text{adj}(A^3))$

• Concept Used: Determinant of the adjoint matrix, $\det(\text{adj}(M)) = (\det(M))^{n-1}$.
• Why: We need to find the determinant of $\text{adj}(A^3)$. Here, the matrix $M$ is $A^3$, and the order $n=3$.
• Calculation: Using the result from Step 1, $\det(A^3) = 8$. $\det(\text{adj}(A^3)) = (\det(A^3))^{n-1} = (8)^{3-1} = 8^2 = 64$ So, $\det(\text{adj}(A^3)) = 64$.

Step 3: Evaluate $\det(5 \text{adj}(A^3))$

• Concept Used: Determinant of a scalar multiple, $\det(kM) = k^n \det(M)$.
• Why: We are now evaluating the determinant of a scalar multiple (5) of the matrix $\text{adj}(A^3)$. Here, the scalar $k=5$, the matrix $M = \text{adj}(A^3)$, and the order $n=3$.
• Calculation: Using the result from Step 2, $\det(\text{adj}(A^3)) = 64$. $\det(5 \text{adj}(A^3)) = 5^n \det(\text{adj}(A^3)) = 5^3 \times 64$ $\det(5 \text{adj}(A^3)) = 125 \times 64$ (We'll keep this in factored form for easier simplification later.)

Step 4: Evaluate $\det(\text{adj}(5 \text{adj}(A^3)))$

• Concept Used: Determinant of the adjoint matrix, $\det(\text{adj}(M)) = (\det(M))^{n-1}$.
• Why: We are again taking the determinant of an adjoint matrix. This time, the matrix $M$ is $5 \text{adj}(A^3)$, and the order $n=3$.
• Calculation: Using the result from Step 3, $\det(5 \text{adj}(A^3)) = 125 \times 64$. $\det(\text{adj}(5 \text{adj}(A^3))) = (\det(5 \text{adj}(A^3)))^{n-1} = (125 \times 64)^{3-1} = (125 \times 64)^2$ So, $\det(\text{adj}(5 \text{adj}(A^3))) = (125 \times 64)^2$.

Step 5: Evaluate the final expression $\det (2 \text{adj} (5 \text{adj} (A^3)))$

• Concept Used: Determinant of a scalar multiple, $\det(kM) = k^n \det(M)$.
• Why: This is the outermost operation, where the entire preceding result is multiplied by the scalar 2. Here, the scalar $k=2$, the matrix $M = \text{adj}(5 \text{adj}(A^3))$, and the order $n=3$.
• Calculation: Using the result from Step 4, $\det(\text{adj}(5 \text{adj}(A^3))) = (125 \times 64)^2$. $\det (2 \text{adj} (5 \text{adj} (A^3))) = 2^n \det (\text{adj} (5 \text{adj} (A^3)))$ $= 2^3 \times (125 \times 64)^2$ $= 8 \times (125 \times 64)^2$

Step 6: Simplify the numerical result

Now we perform the final numerical calculation and express it in a form that matches the given options. $8 \times (125 \times 64)^2$ Let's express the numbers as powers of their prime factors: $8 = 2^3$ $125 = 5^3$ $64 = 2^6$

Substitute these into the expression: $= 2^3 \times (5^3 \times 2^6)^2$ Apply the exponent rules $(ab)^c = a^c b^c$ and $(a^b)^c = a^{bc}$: $= 2^3 \times (5^3)^2 \times (2^6)^2$ $= 2^3 \times 5^6 \times 2^{12}$ Combine the powers of 2: $= 2^{3+12} \times 5^6$ $= 2^{15} \times 5^6$ To match the options which are in the format $X \times 10^Y$, we need to group terms to form powers of $10 = 2 \times 5$. We have $5^6$, so we need $2^6$ to pair with it. $= 2^9 \times (2^6 \times 5^6)$ $= 2^9 \times (2 \times 5)^6$ $= 2^9 \times 10^6$ Finally, calculate $2^9$: $2^9 = 512$ Thus, the final result is: $512 \times 10^6$

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Important Tips and Common Mistakes

• Order of Operations: Always simplify from the innermost part of the expression outwards. This methodical approach helps in correctly applying properties and avoiding errors.
• Scalar Multiplication: A common mistake is to write $\det(kM) = k \det(M)$. Remember the correct formula is $\det(kM) = k^n \det(M)$, where $n$ is the order of the matrix. The exponent $n$ is crucial.
• Adjoint Determinant Exponent: Ensure you use $(n-1)$ as the exponent for $\det(\text{adj}(M)) = (\det(M))^{n-1}$, not $n$.
• Exponent Rules: Be careful with basic exponent rules, especially when raising a power to another power, e.g., $(a^b)^c = a^{bc}$.
• Prime Factorization: For numerical simplification, breaking down numbers into their prime factors (e.g., $125=5^3$, $64=2^6$) can make calculations easier and help in converting the final answer to powers of 10.

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Summary and Key Takeaway

This problem effectively tests a student's ability to apply fundamental properties of determinants and adjoint matrices in a layered manner. The key to success is a clear understanding of each property and its correct application at every step, working systematically from the inside out. Mastering these core properties is essential for tackling more complex problems in matrices and determinants.

The final answer is \boxed{\text{512 \times 10^6}}.