Question

If the matrix \[ \begin{bmatrix} 1 & 12 & 4y \\ 6x & 5 & 2x \\ 8x & 4 & 6 \end{bmatrix} \]
is a symmetric matrix, then the value of \( 2x + y \) is:

• A. -8
• B. 0
• C. 6
• D. 8

Hint:

For a symmetric matrix \( A \), use the condition \( A_{ij} = A_{ji} \) to compare and solve for unknowns. Always match elements across the main diagonal.

Answer 1

The Correct Option is D

Step 1: Understanding Symmetric Matrices 

A matrix \( A \) is symmetric if: \( A^T = A \)

This means that elements across the main diagonal must be equal: \( A_{ij} = A_{ji} \)

Step 2: Equating Corresponding Elements

The given matrix is:

\[ \begin{bmatrix} 1 & 12 & 4y \\ 6x & 5 & 2x \\ 8x & 4 & 6 \end{bmatrix} \]

Since it's symmetric, we equate the off-diagonal elements:

1. \( A_{12} = A_{21} \Rightarrow 12 = 6x \Rightarrow x = \frac{12}{6} = 2 \)
2. \( A_{13} = A_{31} \Rightarrow 4y = 8x \)
   Substituting \( x = 2 \):
   \( 4y = 8 \times 2 = 16 \Rightarrow y = \frac{16}{4} = 4 \)

Step 3: Compute \( 2x + y \)

\( 2x + y = 2(2) + 4 = 4 + 4 = \boxed{8} \)