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: Definition of a Symmetric Matrix

A matrix \( A \) is defined as symmetric when its transpose equals itself: \( A^T = A \). This implies that elements mirrored across the main diagonal are identical: \( A_{ij} = A_{ji} \).

Step 2: Solving for Unknowns

Given the matrix:

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

Due to symmetry, we equate the corresponding off-diagonal elements:

1. Equating \( A_{12} \) and \( A_{21} \): \( 12 = 6x \). Solving for \( x \) yields \( x = \frac{12}{6} = 2 \).
2. Equating \( A_{13} \) and \( A_{31} \): \( 4y = 8x \). Substituting the value of \( x = 2 \) gives \( 4y = 8 \times 2 = 16 \). Solving for \( y \) results in \( y = \frac{16}{4} = 4 \).

Step 3: Calculation of \( 2x + y \)

The expression \( 2x + y \) is calculated as: \( 2(2) + 4 = 4 + 4 = \boxed{8} \).