1. Key Concepts and Formulas

• Matrix Transpose ($A^T$): The transpose of a matrix $A$ is obtained by interchanging its rows and columns.
• Symmetric Matrix: A square matrix $P$ is symmetric if it is equal to its transpose, i.e., $P^T = P$.
• Skew-Symmetric Matrix: A square matrix $Q$ is skew-symmetric if it is equal to the negative of its transpose, i.e., $Q^T = -Q$. For a $2 \times 2$ skew-symmetric matrix, its diagonal elements must be zero, and $Q_{12} = -Q_{21}$.
• Matrix Decomposition: Any square matrix $A$ can be uniquely expressed as the sum of a symmetric matrix $P$ and a skew-symmetric matrix $Q$. The formulas for $P$ and $Q$ are:
  • Symmetric part: $P = \frac{1}{2}(A + A^T)$
  • Skew-symmetric part: $Q = \frac{1}{2}(A - A^T)$
• Determinant of a $2 \times 2$ Matrix: For a matrix $\begin{bmatrix} p & q \\ r & s \end{bmatrix}$, its determinant is $\det = ps - qr$.

2. Step-by-Step Solution

Step 1: Identify the given matrix and its transpose. We are given the matrix $A$: $A = \begin{bmatrix} 2 & 3 \\ a & 0 \end{bmatrix}$ To find the symmetric and skew-symmetric components, we first need to determine the transpose of $A$, denoted as $A^T$. We obtain $A^T$ by swapping the rows and columns of $A$. $A^T = \begin{bmatrix} 2 & a \\ 3 & 0 \end{bmatrix}$ This step is fundamental as $A^T$ is used in the formulas for both $P$ and $Q$.

Step 2: Formulate the skew-symmetric matrix $Q$. According to the key concepts, the skew-symmetric part $Q$ is given by the formula $Q = \frac{1}{2}(A - A^T)$. We need to find $Q$ because its determinant is provided in the problem statement. First, calculate the difference $A - A^T$: $A - A^T = \begin{bmatrix} 2 & 3 \\ a & 0 \end{bmatrix} - \begin{bmatrix} 2 & a \\ 3 & 0 \end{bmatrix} = \begin{bmatrix} 2-2 & 3-a \\ a-3 & 0-0 \end{bmatrix} = \begin{bmatrix} 0 & 3-a \\ a-3 & 0 \end{bmatrix}$ Now, multiply by $\frac{1}{2}$ to get $Q$: $Q = \frac{1}{2} \begin{bmatrix} 0 & 3-a \\ a-3 & 0 \end{bmatrix} = \begin{bmatrix} 0 & \frac{3-a}{2} \\ \frac{a-3}{2} & 0 \end{bmatrix}$ We can verify that $Q$ is indeed skew-symmetric, as its diagonal elements are zero and $Q_{12} = -Q_{21}$ (since $\frac{3-a}{2} = -\frac{a-3}{2}$).

Step 3: Use the given determinant of $Q$ to find the possible values of $a$. We are given that $\det(Q) = 9$. We will use the $2 \times 2$ determinant formula to calculate $\det(Q)$ and then equate it to 9. For $Q = \begin{bmatrix} 0 & \frac{3-a}{2} \\ \frac{a-3}{2} & 0 \end{bmatrix}$, the determinant is: $\det(Q) = (0)(0) - \left(\frac{3-a}{2}\right)\left(\frac{a-3}{2}\right)$ $\det(Q) = - \frac{(3-a)(a-3)}{4}$ Since $(3-a) = -(a-3)$, we can rewrite the expression as: $\det(Q) = - \frac{-(a-3)(a-3)}{4} = \frac{(a-3)^2}{4}$ Now, set this equal to the given value $\det(Q) = 9$: $\frac{(a-3)^2}{4} = 9$ Multiply both sides by 4: $(a-3)^2 = 36$ Take the square root of both sides. It's crucial to remember both positive and negative roots: $a-3 = \pm \sqrt{36}$ $a-3 = \pm 6$ This yields two possible values for $a$:

• Case 1: $a-3 = 6 \implies a = 6+3 = 9$
• Case 2: $a-3 = -6 \implies a = -6+3 = -3$ These values of $a$ are critical because they will determine the specific form and determinant of matrix $P$.

Step 4: Formulate the symmetric matrix $P$. The symmetric part $P$ is given by the formula $P = \frac{1}{2}(A + A^T)$. We need to find $P$ to calculate its determinant as required by the question. First, calculate the sum $A + A^T$: $A + A^T = \begin{bmatrix} 2 & 3 \\ a & 0 \end{bmatrix} + \begin{bmatrix} 2 & a \\ 3 & 0 \end{bmatrix} = \begin{bmatrix} 2+2 & 3+a \\ a+3 & 0+0 \end{bmatrix} = \begin{bmatrix} 4 & 3+a \\ a+3 & 0 \end{bmatrix}$ Now, multiply by $\frac{1}{2}$ to get $P$: $P = \frac{1}{2} \begin{bmatrix} 4 & 3+a \\ a+3 & 0 \end{bmatrix} = \begin{bmatrix} 2 & \frac{a+3}{2} \\ \frac{a+3}{2} & 0 \end{bmatrix}$ We can verify that $P$ is indeed symmetric, as $P_{12} = P_{21}$.

Step 5: Calculate the determinant of $P$ for each possible value of $a$. The determinant of $P$ is calculated using the $2 \times 2$ determinant formula: $\det(P) = (2)(0) - \left(\frac{a+3}{2}\right)\left(\frac{a+3}{2}\right) = 0 - \frac{(a+3)^2}{4} = - \frac{(a+3)^2}{4}$ Now, we evaluate $\det(P)$ for each of the possible values of $a$ found in Step 3:

• For $a = 9$: Substitute $a=9$ into the expression for $\det(P)$: $\det(P_1) = - \frac{(9+3)^2}{4} = - \frac{(12)^2}{4} = - \frac{144}{4} = -36$

• For $a = -3$: Substitute $a=-3$ into the expression for $\det(P)$: $\det(P_2) = - \frac{(-3+3)^2}{4} = - \frac{(0)^2}{4} = 0$ These are the two possible values for the determinant of $P$.

Step 6: Find the modulus of the sum of all possible values of $\det(P)$. The problem asks for the modulus of the sum of all possible values of $\det(P)$. The possible values for $\det(P)$ are $-36$ and $0$. First, calculate their sum: $S = -36 + 0 = -36$ Next, find the modulus of this sum: $|S| = |-36| = 36$

3. Common Mistakes & Tips

• Algebraic Precision: Be extremely careful with arithmetic, especially when dealing with squares and square roots. Forgetting the $\pm$ in $\sqrt{36}$ is a common error that leads to missing a possible value for $a$.
• Definition Check: Always quickly verify if the matrices $P$ and $Q$ you've calculated indeed satisfy the symmetric ($P^T=P$) and skew-symmetric ($Q^T=-Q$) properties, respectively. This can catch calculation errors early.
• Read the Question Carefully: Distinguish between "sum of moduli" and "modulus of the sum". The question explicitly asks for the latter.
• Determinant Formula: Ensure correct application of the determinant formula for $2 \times 2$ matrices.

4. Summary

This problem required us to apply the fundamental concept of decomposing a square matrix into its unique symmetric and skew-symmetric parts. We first calculated the skew-symmetric matrix $Q$ using the given matrix $A$ and its transpose. By utilizing the provided determinant of $Q$, we solved for the unknown parameter $a$, which yielded two possible values. Subsequently, we formulated the symmetric matrix $P$ and calculated its determinant for each of the possible values of $a$. Finally, we summed these determinant values and took the modulus of the sum to arrive at the required answer. The process highlights the importance of systematic application of matrix properties and careful algebraic manipulation.

5. Final Answer

The modulus of the sum of all possible values of determinant of P is 36. The final answer is $\boxed{\text{36}}$, which corresponds to option (A).