Question

A matrix whose elements \( a_{ij} \) are defined by \( a_{ij} = \frac{1}{3}|i - 5j|, \; i, j = 1, 2, 3 \) is

• A. \( \begin{bmatrix} 4 & 3 & \frac{14}{3} \\ 1 & \frac{8}{3} & 13 \\ \frac{2}{3} & \frac{7}{3} & 4 \end{bmatrix} \)
• B. \( \begin{bmatrix} 4 & \frac{3}{3} & \frac{14}{3} \\ 1 & \frac{8}{3} & \frac{13}{3} \\ \frac{2}{3} & \frac{7}{3} & 4 \end{bmatrix} \)
• C. \( \begin{bmatrix} 4 & 3 & \frac{14}{3} \\ 1 & \frac{8}{3} & \frac{13}{3} \\ \frac{2}{3} & \frac{7}{3} & 4 \end{bmatrix} \)
• D. \( \begin{bmatrix} 4 & 3 & 10 \\ 1 & 8 & 13 \\ 2 & 7 & 4\\ \end{bmatrix} \)

Hint:

Always evaluate the absolute value expression first before dividing. Use exact fractions if given.

Answer 1

The Correct Option is B

Step 1: Use the formula given.
We are given \( a_{ij} = \frac{1}{3}|i - 5j| \) for \( i, j = 1, 2, 3 \). Compute each element. \[ \begin{aligned} a_{11} &= \frac{1}{3}|1 - 5(1)| = \frac{1}{3}|-4| = \frac{4}{3} \\ a_{12} &= \frac{1}{3}|1 - 5(2)| = \frac{1}{3}|-9| = 3 \\ a_{13} &= \frac{1}{3}|1 - 5(3)| = \frac{1}{3}|-14| = \frac{14}{3} \\ \text{(Continue for all entries...) } \end{aligned} \] After full computation, we match with the matrix in option (2).