2.2 Q-1

Exercise 2.2 — Question 1

Constructing a Matrix from a Formula

In this type of question, a matrix $A=[a_{ij}]$ of a given order $m\times n$ is constructed by substituting values of $i$ (row index) and $j$ (column index) into a given formula for the general element $a_{ij}$.

Key notation:

• $i$ = row number, $1\le i\le m$
• $j$ = column number, $1\le j\le n$
• $a_{ij}$ = element at row $i$, column $j$

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Method

1. Identify the order of the matrix (e.g., $2\times 3$ means 2 rows, 3 columns).
2. For each position $(i,j)$, substitute the values into the formula to find $a_{ij}$.
3. Arrange the computed values into the matrix.

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Worked Example

Construct a $2\times 3$ matrix $A=[a_{ij}]$ where $a_{ij}=2i-j$.

The matrix has $m=2$ rows and $n=3$ columns. Compute each element:

So: $A=\left[\begin{array}{ccc}1& 0& -1\\ 3& 2& 1\end{array}\right]$

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Another Example with Complex/Fractional Formula

Construct a $2\times 2$ matrix $B=[b_{ij}]$ where $b_{ij}=\frac{i^2-j}{3}$.

Compute each element:

$b_{11}=\frac{1-1}{3}=0,b_{12}=\frac{1-2}{3}=-\frac{1}{3}$

$b_{21}=\frac{4-1}{3}=1,b_{22}=\frac{4-2}{3}=\frac{2}{3}$

So: $B=\left[\begin{array}{cc}0& -\frac{1}{3}\\ 1& \frac{2}{3}\end{array}\right]$

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