Q.Construct a $3 \times 2$ matrix whose elements are given by $a_{ij} = \frac{1}{2}\,|\,i - 3j\,|$.

Concept: Matrix Construction — each element is defined by a formula in terms of row index $i$ and column index $j$.

Step 1: The matrix is $3 \times 2$, so $i = 1,2,3$ and $j = 1,2$.

Step 2: Compute each $a_{ij} = \frac{1}{2} |i - 3j|$.

• For $j=1$:

  $a_{11} = \frac{1}{2}|1-3| = 1$,

  $a_{21} = \frac{1}{2}|2-3| = \frac{1}{2}$,

  $a_{31} = \frac{1}{2}|3-3| = 0$.

• For $j=2$:

  $a_{12} = \frac{1}{2}|1-6| = \frac{5}{2}$,

  $a_{22} = \frac{1}{2}|2-6| = 2$, …

The core idea here is straightforward: a matrix is just a rectangular array of numbers, and each entry has a specific address — row $i$, column $j$. The formula $a_{ij} = \frac{1}{2}|i - 3j|$ tells us exactly how to compute the number at that address. The absolute value ensures every entry is non-negative, and the factor $\frac{1}{2}$ scales the result.

Let’s walk through it systematically.

1. Identify the range of indices.

   The matrix is $3 \times 2$, so $i$ runs from 1 to 3 (rows), and $j$ runs from 1 to 2 (columns). That gives us $3 \times 2 = 6$ entries to compute.

2. Compute each entry using the formula.

   For each pair $(i, j)$, we evaluate $a_{ij} = \frac{1}{2}|i - 3j|$.

   Let’s do it row by row.

   Row 1 ($i = 1$):

   • $j = 1$: $a_{11} = \frac{1}{2}|1 - 3(1)| = \frac{1}{2}|1 - 3| = \frac{1}{2}|-2| = \frac{1}{2} \times 2 = 1$
   • $j = 2$: $a_{12} = \frac{1}{2}|1 - 3(2)| = \frac{1}{2}|1 - 6| = \frac{1}{2}|-5| = \frac{1}{2} \times 5 = \frac{5}{2}$

   Row 2 ($i = 2$):

   • $j = 1$: $a_{21} = \frac{1}{2}|2 - 3(1)| = \frac{1}{2}|2 - 3| = \frac{1}{2}|-1| = \frac{1}{2} \times 1 = \frac{1}{2}$
   • $j = 2$: $a_{22} = \frac{1}{2}|2 - 3(2)| = \frac{1}{2}|2 - 6| = \frac{1}{2}|-4| = \frac{1}{2} \times 4 = 2$

   Row 3 ($i = 3$):

   • $j = 1$: $a_{31} = \frac{1}{2}|3 - 3(1)| = \frac{1}{2}|3 - 3| = \frac{1}{2} \times 0 = 0$
   • $j = 2$: $a_{32} = \frac{1}{2}|3 - 3(2)| = \frac{1}{2}|3 - 6| = \frac{1}{2}|-3| = \frac{1}{2} \times 3 = \frac{3}{2}$

3. Assemble the matrix. …

Method: Constructing a matrix from a formula $a_{ij} = f(i,j)$

Use this whenever a matrix of given order is defined by a rule for its general element.

Steps

Step 1: Fix the index ranges from the order.

For an $m\times n$ matrix, $i = 1,\dots,m$ (rows) and $j = 1,\dots,n$ (columns).

Step 2: Substitute each $(i,j)$ into the formula. …

Common Mistakes

Mistake 1: Dropping the absolute value.

Why it's wrong: $\tfrac12|i-3j|$ is always non-negative; ignoring $|\cdot|$ produces wrong negative entries like $-\tfrac52$. Correct approach: take the modulus before scaling.

Mistake 2: Swapping the roles of $i$ and $j$. …