Question

Consider a 3 x 3 matrix A whose \( (i, j) \)-th element, \( a_{i,j} = (i - j)^3 \). Then the matrix A will be

• A. symmetric.
• B. skew-symmetric.
• C. unitary.
• D. null.

Hint:

A matrix is skew-symmetric if \( A = -A^T \), meaning each off-diagonal element is the negative of its transpose counterpart.

Answer 1

The Correct Option is B

Step 1: Understanding the matrix elements.
The given matrix \( A \) has elements defined by \( a_{i,j} = (i - j)^3 \). Let's compute the elements for a 3 x 3 matrix: \[ A = \begin{pmatrix} a_{1,1} & a_{1,2} & a_{1,3} \\ a_{2,1} & a_{2,2} & a_{2,3} \\ a_{3,1} & a_{3,2} & a_{3,3} \end{pmatrix} = \begin{pmatrix} 0 & -1 & -8 \\ 1 & 0 & -1 \\ 8 & 1 & 0 \end{pmatrix}. \] Step 2: Checking the properties of the matrix.
To check if the matrix is symmetric or skew-symmetric, we verify if \( A = A^T \) (symmetric) or \( A = -A^T \) (skew-symmetric). Upon transposing the matrix, we observe that: \[ A^T = \begin{pmatrix} 0 & 1 & 8 \\ -1 & 0 & 1 \\ -8 & -1 & 0 \end{pmatrix}. \] Since \( A = -A^T \), the matrix is skew-symmetric. Step 3: Conclusion.
The matrix is skew-symmetric, so the correct answer is (B).