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 symmetric matrices always remember: \(a_{ij}=a_{ji}\). So elements across the diagonal must be equal.

Answer 1

The Correct Option is A

Step 1: Recall the definition of symmetric matrix.
A matrix is called symmetric if
\[ A = A^T \] where \(A^T\) is the transpose of matrix \(A\).
This means
\[ a_{ij} = a_{ji} \] for every element of the matrix.
Step 2: Compare corresponding elements.
Given matrix
\[ A = \begin{bmatrix} 1 & a & b \\ -1 & 2 & c \\ 0 & 5 & 3 \end{bmatrix} \] For symmetry we compare elements across the main diagonal.
First pair:
\[ a_{12} = a_{21} \] \[ a = -1 \] Second pair:
\[ a_{13} = a_{31} \] \[ b = 0 \] Third pair:
\[ a_{23} = a_{32} \] \[ c = 5 \] Step 3: Substitute values.
Now compute
\[ 3a + b + c \] Substitute values
\[ 3(-1) + 0 + 5 \] \[ = -3 + 5 \] \[ = 2 \] Step 4: Final conclusion.
Therefore the required value becomes
\[ 3a + b + c = 2 \] Final Answer: \( \boxed{2} \)