Key Concepts and Formulas

This problem can be efficiently solved by leveraging fundamental properties of determinants. The key properties we will use are:

1. Determinant of a Product: For any two square matrices $M$ and $N$ of the same order, the determinant of their product is the product of their individual determinants: $\operatorname{det}(MN) = \operatorname{det}(M) \operatorname{det}(N)$ This property extends to any number of matrices: $\operatorname{det}(M_1 M_2 \dots M_k) = \operatorname{det}(M_1) \operatorname{det}(M_2) \dots \operatorname{det}(M_k)$.

2. Determinant of a Transpose: The determinant of a matrix is equal to the determinant of its transpose: $\operatorname{det}(M^T) = \operatorname{det}(M)$

3. Determinant of a Power: For any positive integer $k$, the determinant of a matrix raised to the power $k$ is the $k$-th power of its determinant: $\operatorname{det}(M^k) = (\operatorname{det}(M))^k$

4. Determinant of a $2 \times 2$ Matrix: For a $2 \times 2$ matrix $\left[\begin{array}{ll} a & b \\ c & d \end{array}\right]$, its determinant is calculated as $ad - bc$.

Step-by-Step Solution

Step 1: Calculate the Determinants of Matrices A and B We begin by calculating the determinants of the given matrices $A$ and $B$. This is a crucial first step because all subsequent calculations will rely on these fundamental values, using the determinant properties.

Given matrix $A$: $A=\left[\begin{array}{cc} \sqrt{2} & 1 \\ -1 & \sqrt{2} \end{array}\right]$ Using the $2 \times 2$ determinant formula $ad - bc$: $\operatorname{det}(A) = (\sqrt{2})(\sqrt{2}) - (1)(-1) = 2 - (-1) = 2 + 1 = 3$

Given matrix $B$: $B=\left[\begin{array}{ll} 1 & 0 \\ 1 & 1 \end{array}\right]$ Using the $2 \times 2$ determinant formula $ad - bc$: $\operatorname{det}(B) = (1)(1) - (0)(1) = 1 - 0 = 1$

Step 2: Express $\operatorname{det}(C)$ in terms of $\operatorname{det}(A)$ and $\operatorname{det}(B)$ Next, we need to find the determinant of matrix $C$. Directly performing the matrix multiplication $ABA^T$ would be lengthy and prone to errors. Instead, we strategically use the determinant properties to simplify this task.

We are given $C = ABA^T$. Applying the determinant of a product property, $\operatorname{det}(M_1 M_2 M_3) = \operatorname{det}(M_1) \operatorname{det}(M_2) \operatorname{det}(M_3)$: $\operatorname{det}(C) = \operatorname{det}(ABA^T) = \operatorname{det}(A) \operatorname{det}(B) \operatorname{det}(A^T)$ Now, applying the determinant of a transpose property, $\operatorname{det}(A^T) = \operatorname{det}(A)$: $\operatorname{det}(C) = \operatorname{det}(A) \operatorname{det}(B) \operatorname{det}(A)$ Simplifying this expression: $\operatorname{det}(C) = (\operatorname{det}(A))^2 \operatorname{det}(B)$

Step 3: Express $\operatorname{det}(X)$ in terms of $\operatorname{det}(A)$ and $\operatorname{det}(C)$ Similarly, to find $\operatorname{det}(X)$, we avoid direct matrix computation of $X = A^T C^2 A$. This would involve squaring matrix $C$ (which is itself a product) and then performing further matrix multiplications, which is highly inefficient. We once again leverage determinant properties.

We are given $X = A^T C^2 A$. Applying the determinant of a product property: $\operatorname{det}(X) = \operatorname{det}(A^T C^2 A) = \operatorname{det}(A^T) \operatorname{det}(C^2) \operatorname{det}(A)$ Now, applying the determinant of a transpose property, $\operatorname{det}(A^T) = \operatorname{det}(A)$, and the determinant of a power property, $\operatorname{det}(C^2) = (\operatorname{det}(C))^2$: $\operatorname{det}(X) = \operatorname{det}(A) (\operatorname{det}(C))^2 \operatorname{det}(A)$ Simplifying this expression: $\operatorname{det}(X) = (\operatorname{det}(A))^2 (\operatorname{det}(C))^2$

Step 4: Substitute and Simplify the Expression for $\operatorname{det}(X)$ At this stage, we have expressions for $\operatorname{det}(C)$ and $\operatorname{det}(X)$ in terms of other determinants. To find the numerical value of $\operatorname{det}(X)$, we substitute the expression for $\operatorname{det}(C)$ (from Step 2) into the expression for $\operatorname{det}(X)$ (from Step 3). This step allows us to express $\operatorname{det}(X)$ purely in terms of $\operatorname{det}(A)$ and $\operatorname{det}(B)$, whose numerical values we have already calculated.

From Step 2: $\operatorname{det}(C) = (\operatorname{det}(A))^2 \operatorname{det}(B)$ From Step 3: $\operatorname{det}(X) = (\operatorname{det}(A))^2 (\operatorname{det}(C))^2$

Substitute the expression for $\operatorname{det}(C)$ into the expression for $\operatorname{det}(X)$: $\operatorname{det}(X) = (\operatorname{det}(A))^2 \left( (\operatorname{det}(A))^2 \operatorname{det}(B) \right)^2$ Now, carefully simplify the expression using exponent rules. Remember that $(uv)^k = u^k v^k$: $\operatorname{det}(X) = (\operatorname{det}(A))^2 \left( (\operatorname{det}(A))^{2 \times 2} (\operatorname{det}(B))^2 \right)$ $\operatorname{det}(X) = (\operatorname{det}(A))^2 \left( (\operatorname{det}(A))^4 (\operatorname{det}(B))^2 \right)$ Combine the powers of $\operatorname{det}(A)$ (using $a^m a^n = a^{m+n}$): $\operatorname{det}(X) = (\operatorname{det}(A))^{2+4} (\operatorname{det}(B))^2$ $\operatorname{det}(X) = (\operatorname{det}(A))^6 (\operatorname{det}(B))^2$

Step 5: Final Calculation Finally, we substitute the numerical values of $\operatorname{det}(A)$ and $\operatorname{det}(B)$ that we calculated in Step 1 into our simplified expression for $\operatorname{det}(X)$.

We found: $\operatorname{det}(A) = 3$ $\operatorname{det}(B) = 1$

Substituting these values into $\operatorname{det}(X) = (\operatorname{det}(A))^6 (\operatorname{det}(B))^2$: $\operatorname{det}(X) = (3)^6 (1)^2$ $\operatorname{det}(X) = 729 \times 1$ $\operatorname{det}(X) = 729$

Common Mistakes & Tips

• Incorrect Exponent Distribution: A common mistake is to incorrectly distribute the square in the expression $(\operatorname{det}(A)^2 \operatorname{det}(B))^2$. Remember that $(uv)^k = u^k v^k$, so $((\operatorname{det}(A))^2 \operatorname{det}(B))^2 = ((\operatorname{det}(A))^2)^2 (\operatorname{det}(B))^2 = (\operatorname{det}(A))^4 (\operatorname{det}(B))^2$.
• Avoid Direct Matrix Multiplication: The problem is designed to test your understanding of determinant properties. Attempting to perform actual matrix multiplications for $C$ and $X$ would be extremely time-consuming and prone to errors. Always look for opportunities to use determinant properties for simplification.
• Careful with Transpose Property: Ensure you correctly apply $\operatorname{det}(M^T) = \operatorname{det}(M)$. It's a fundamental property that simplifies expressions involving transposes.

Summary

This problem effectively demonstrates the power and efficiency of using fundamental determinant properties to solve complex matrix problems. By strategically applying the properties of determinants for products, transposes, and powers, we were able to express $\operatorname{det}(X)$ solely in terms of $\operatorname{det}(A)$ and $\operatorname{det}(B)$, thus avoiding any cumbersome matrix multiplications. The final calculation involved simple exponentiation, yielding $\operatorname{det}(X) = 729$.

The final answer is $\boxed{\text{729}}$, which corresponds to option (B).