Q.Construct a $2 \times 2$ matrix, $A = [a_{ij}]$, whose elements are given by:

Substitute the row index $i$ and column index $j$ (each $1$ or $2$) into each rule.

(i) $a_{ij}=\frac{(i+j)^2}{2}$: $a_{11}=\frac{4}{2}=2,\ a_{12}=a_{21}=\frac{9}{2},\ a_{22}=\frac{16}{2}=8.$

(ii) $a_{ij}=\frac{i}{j}$: $a_{11}=1,\ a_{12}=\frac{1}{2},\ a_{21}=2,\ a_{22}=1.$

(iii) $a_{ij}=\frac{(i+2j)^2}{2}$: $a_{11}=\frac{9}{2},\ a_{12}=\frac{25}{2},\ a_{21}=8,\ a_{22}=18.$ …

A $2\times2$ matrix has rows $i=1,2$ and columns $j=1,2$. For each of the four positions, substitute the row number $i$ and column number $j$ into the given formula — careful arithmetic, no hidden trick.

(i) $a_{ij}=\dfrac{(i+j)^2}{2}$

• $a_{11}=\frac{(1+1)^2}{2}=\frac{4}{2}=2$
• $a_{12}=\frac{(1+2)^2}{2}=\frac{9}{2}$
• $a_{21}=\frac{(2+1)^2}{2}=\frac{9}{2}$
• $a_{22}=\frac{(2+2)^2}{2}=\frac{16}{2}=8$

It comes out symmetric because $(i+j)^2$ is symmetric in $i$ and $j$.

(ii) $a_{ij}=\dfrac{i}{j}$

• $a_{11}=\frac{1}{1}=1,\qquad a_{12}=\frac{1}{2}$
• $a_{21}=\frac{2}{1}=2,\qquad a_{22}=\frac{2}{2}=1$

Keep $i$ as the row and $j$ as the column: $a_{21}=\frac{2}{1}=2$, not $\frac{1}{2}$.

(iii) $a_{ij}=\dfrac{(i+2j)^2}{2}$

• $a_{11}=\frac{(1+2)^2}{2}=\frac{9}{2}$ …

Method: Constructing a Matrix From an Element Formula $a_{ij}$

Use this whenever a matrix is defined by a rule for its general element $a_{ij}$. Systematically substitute each valid $(i,j)$ into the rule.

Steps

Step 1: Fix the ranges of the indices from the required order.

For a $2\times2$ matrix, $i$ (row) runs over $1,2$ and $j$ (column) runs over $1,2$, giving four positions to fill. Keep the convention: $i$ is the row, $j$ is the column.

Step 2: Substitute each $(i,j)$ into the formula, one entry at a time.

Compute $a_{11},a_{12},a_{21},a_{22}$ by plugging the numbers into the given expression, doing the arithmetic carefully (squares, fractions, absolute values as they appear). …

Common Mistakes

Mistake 1: Swapping the row and column indices.

Why it's wrong: $a_{21}$ uses $i=2,\,j=1$; for a rule like $a_{ij}=i/j$ that gives $2/1=2$, not $1/2$. Correct approach: always read $i$ as the row and $j$ as the column, in that order.

Mistake 2: Assuming the matrix must be symmetric. …