Question

If \( A = \begin{bmatrix} 1 & a & b \\ -1 & 2 & c \\ 0 & 5 & 3 \end{bmatrix} \) is a symmetric matrix, then the value of \( 3a + b + c \) is

• A. \( 2 \)
• B. \( 6 \)
• C. \( 4 \)
• D. \( 0 \)

Hint:

For symmetry, just "reflect" the elements across the diagonal to find unknowns.
Remember that the diagonal elements themselves don't provide any information about symmetry constraints.

Answer 1

The Correct Option is A

Concept:

• A square matrix \( A \) is symmetric if \( A = A^T \), where \( A^T \) is the transpose of \( A \).
• For a symmetric matrix, elements across the main diagonal are equal: \( a_{ij} = a_{ji} \).

Step 1: Apply the condition for a symmetric matrix
Given \( A = \begin{bmatrix} 1 & a & b \\ -1 & 2 & c \\ 0 & 5 & 3 \end{bmatrix} \).
The transpose is \( A^T = \begin{bmatrix} 1 & -1 & 0 \\ a & 2 & 5 \\ b & c & 3 \end{bmatrix} \).
Since \( A = A^T \), we equate corresponding elements:
• \( a_{12} = a_{21} \implies a = -1 \)
• \( a_{13} = a_{31} \implies b = 0 \)
• \( a_{23} = a_{32} \implies c = 5 \)

Step 2: Calculate the required expression
We need to find the value of \( 3a + b + c \).
Substitute the values found in
Step 1:
\[ 3a + b + c = 3(-1) + 0 + 5 \]
\[ = -3 + 5 \]
\[ = 2 \]