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

To determine the value of \(3a + b + c\), we begin by analyzing the properties of a symmetric matrix. A matrix is symmetric if it is equal to its transpose. Therefore, for the matrix \(A\) to be symmetric, the condition \(A = A^T\) must hold. 

The given matrix is:

\(A\)

=

\(\begin{bmatrix} 1 & a & b \\ -1 & 2 & c \\ 0 & 5 & 3 \end{bmatrix}\)

The transpose of matrix \(A\) is:

\(A^T\)

=

\(\begin{bmatrix} 1 & -1 & 0 \\ a & 2 & 5 \\ b & c & 3 \end{bmatrix}\)

For \(A\) to be symmetric, \(A = A^T\) must be true, implying:

1. The element in the first row and second column must be equal to the element in the second row and first column: \(a = -1\).
2. The element in the first row and third column must be equal to the element in the third row and first column: \(b = 0\).
3. The element in the second row and third column must be equal to the element in the third row and second column: \(c = 5\).

Now that we have the values of \(a\), \(b\), and \(c\), we can substitute them into the expression \(3a + b + c\):

\[ 3a + b + c = 3(-1) + 0 + 5 \]

Simplifying this expression gives:

\[ 3(-1) + 0 + 5 = -3 + 5 = 2 \]

Thus, the value of \(3a + b + c\) is 2.