Question

The value of k for which the matrix \(\begin{pmatrix} 0 & 2 & 4 \\ 2 & 0 & 5 \\ -3 & 5 & 0 \end{pmatrix}\) is a symmetric matrix is given by :

• A. 3
• B. -3
• C. 0
• D. 1

Answer 1

The Correct Option is B

To determine the value of \( k \) for which the matrix \(\begin{pmatrix} 0 & 2 & 4 \\ 2 & 0 & 5 \\ -3 & 5 & 0 \end{pmatrix}\) is symmetric, we need to understand the properties of a symmetric matrix.

A symmetric matrix is such that it is equal to its own transpose. Mathematically, a matrix \( A \) is symmetric if \( A = A^T \).

For the given matrix \( A = \begin{pmatrix} 0 & 2 & 4 \\ 2 & 0 & 5 \\ -3 & 5 & 0 \end{pmatrix} \), it is already almost symmetric except for the element in the third row, first column compared with the first row, third column.

The matrix \( A \) should satisfy the condition: \( A_{ij} = A_{ji} \) for all \( i, j \).

Since the corresponding element for \( A_{13} \) and \( A_{31} \) are \( 4 \) and \(-3\) respectively: \[ A_{13} = A_{31} \Rightarrow 4 = -3 \]

Here, it seems like there is a mistake. To make the matrix symmetric, \( A_{31} = A_{13} \), so: \[-3 = k\]

Therefore, the value of \( k \) which makes the matrix symmetric is \( -3 \).