Key Concepts and Formulas

1. Arithmetic Progression (AP): If three numbers $x, y, z$ are in AP, then the middle term $y$ is the arithmetic mean of the other two, meaning $2y = x+z$. This fundamental relationship is key to simplifying the determinant.
2. Determinant Properties:
   • The value of a determinant remains unchanged if we apply elementary row operations of the type $R_i \to R_i + cR_j$ (or more generally, $R_i \to R_i + c_1R_j + c_2R_k$) to any row $R_i$. Similarly, this holds for column operations.
   • If a row or column of a determinant contains only zeros, the determinant's value is zero.
   • To evaluate a $3 \times 3$ determinant: $\begin{vmatrix} a & b & c \\ d & e & f \\ g & h & i \end{vmatrix} = a(ei - fh) - b(di - fg) + c(dh - eg)$ Expanding along a row or column that contains more zeros significantly simplifies the calculation.

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Step-by-Step Solution

Step 1: Write Down the Given Determinant Equation

We are provided with a $3 \times 3$ matrix whose determinant is zero: $\begin{vmatrix} 3 & {4\sqrt 2 } & x \\ 4 & {5\sqrt 2 } & y \\ 5 & k & z \end{vmatrix} = 0$

Step 2: Utilize the Arithmetic Progression Property

We are given that $x, y, z$ are in an Arithmetic Progression. Reasoning: For any three terms in an AP, the sum of the first and third terms is equal to twice the middle term. Therefore, we have the relation: $x+z = 2y$ This equation can be rearranged as $x - 2y + z = 0$. This specific relationship will be instrumental in simplifying the determinant.

Step 3: Perform a Strategic Row Operation

Reasoning: Our goal is to simplify the determinant by creating zeros. Observing the relation $x - 2y + z = 0$ from the AP property, we can apply a row operation to the first row ($R_1$) that combines elements from all three rows to make the third element of the new first row zero. We apply the row operation $R_1 \to R_1 - 2R_2 + R_3$. Let's calculate the new elements for the first row:

• New element at (1,1) position: $3 - 2(4) + 5 = 3 - 8 + 5 = 0$
• New element at (1,2) position: $4\sqrt{2} - 2(5\sqrt{2}) + k = 4\sqrt{2} - 10\sqrt{2} + k = k - 6\sqrt{2}$
• New element at (1,3) position: $x - 2y + z$. Using the AP property $x+z=2y$, we substitute $x+z$ with $2y$: $2y - 2y = 0$.

Now, the determinant transforms into: $\begin{vmatrix} 0 & {k - 6\sqrt 2 } & 0 \\ 4 & {5\sqrt 2 } & y \\ 5 & k & z \end{vmatrix} = 0$

Step 4: Expand the Simplified Determinant

Reasoning: With two zeros in the first row, expanding the determinant along the first row is the most efficient method. The expansion formula along the first row is $a_{11} \cdot C_{11} + a_{12} \cdot C_{12} + a_{13} \cdot C_{13}$, where $C_{ij}$ is the cofactor of element $a_{ij}$. Given $a_{11}=0$, $a_{12}=k-6\sqrt{2}$, and $a_{13}=0$, the expansion simplifies to: $0 \cdot C_{11} + (k - 6\sqrt{2}) \cdot C_{12} + 0 \cdot C_{13} = 0$ This means we only need to calculate the cofactor $C_{12}$. The cofactor $C_{12} = (-1)^{1+2} \cdot \begin{vmatrix} 4 & y \\ 5 & z \end{vmatrix} = - (4z - 5y)$. Substituting this back into the determinant equation: $(k - 6\sqrt{2}) \cdot (-(4z - 5y)) = 0$ Multiplying by $-1$ on both sides to simplify: $(k - 6\sqrt{2})(4z - 5y) = 0$

Step 5: Identify Potential Solutions for k

From the product of two terms being zero, we have two possible scenarios:

1. $k - 6\sqrt{2} = 0 \implies k = 6\sqrt{2}$
2. $4z - 5y = 0 \implies 4z = 5y$

Step 6: Validate Solutions Using Problem Constraints

Reasoning: We must check if the second possibility, $4z = 5y$, is consistent with all the given conditions, especially $x \ne 3d$. Let $d$ be the common difference of the arithmetic progression $x, y, z$. Then we can express $y$ and $z$ in terms of $x$ and $d$: $y = x+d$ $z = x+2d$ Substitute these expressions into the condition $4z = 5y$: $4(x+2d) = 5(x+d)$ $4x + 8d = 5x + 5d$ Rearranging the terms to solve for $x$: $8d - 5d = 5x - 4x$ $3d = x$ Reasoning: The problem statement explicitly provides the constraint $x \ne 3d$. Our derived condition $x=3d$ directly contradicts this given information. Therefore, the possibility $4z - 5y = 0$ is not a valid solution under the problem's constraints.

Step 7: Calculate the Final Value of $k^2$

Since the condition $4z - 5y = 0$ is ruled out by the problem's constraints, the only remaining valid possibility is $k - 6\sqrt{2} = 0$. $k = 6\sqrt{2}$ The question asks for the value of $k^2$: $k^2 = (6\sqrt{2})^2 = 6^2 \cdot (\sqrt{2})^2 = 36 \cdot 2 = 72$

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Common Mistakes & Tips

• Overlooking Constraints: Always pay close attention to all conditions given in the problem statement (e.g., $x \ne 3d$). These constraints are often crucial for eliminating extraneous solutions.
• Incorrect Row Operations: Ensure that the chosen row operation does not change the determinant's value (e.g., $R_i \to cR_i$ changes the determinant by a factor of $c$). Operations of the type $R_i \to R_i + cR_j$ are safe.
• Sign Errors in Cofactors: Be careful with the alternating signs when expanding a determinant. For $C_{12}$, the sign is $(-1)^{1+2} = -1$.
• Recognizing Patterns: For problems involving AP, GP, or HP, look for combinations of terms that can be formed using row/column operations to create zeros or simplify expressions. The $R_1 \to R_1 - 2R_2 + R_3$ operation is a classic trick for AP.

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Summary

This problem effectively combines the properties of Arithmetic Progression with determinant evaluation. The key insight was to leverage the AP property ($x+z=2y$) to perform a strategic row operation ($R_1 \to R_1 - 2R_2 + R_3$) that simplified the determinant by introducing zeros. After expanding the simplified determinant, two potential values for $k$ emerged. The critical constraint $x \ne 3d$ provided in the problem statement was then used to eliminate one of these possibilities, leading to a unique value for $k$ and subsequently $k^2$.

The final answer is $\boxed{72}$, which corresponds to option (A).