2.2 Q-3 | Matrices And Determinants | Mathematics 11

Exercise 2.2 — Question 3

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.

Key Convention:

$i$ = row number, where $1 \leq i \leq m$
$j$ = column number, where $1 \leq j \leq n$
The element $a_{ij}$ occupies row $i$ and column $j$ .

Typical Problem Format

Construct a matrix of order $2 \times 3$ if $a_{ij}$ is defined by a given rule, for example:

$a_{ij} = i + 2 j$

Step-by-step solution:

For a $2 \times 3$ matrix, $i \in {1, 2}$ and $j \in {1, 2, 3}$ .

Compute each element:

	$j = 1$	$j = 2$	$j = 3$
$i = 1$	$a_{11} = 1 + 2 (1) = 3$	$a_{12} = 1 + 2 (2) = 5$	$a_{13} = 1 + 2 (3) = 7$
$i = 2$	$a_{21} = 2 + 2 (1) = 4$	$a_{22} = 2 + 2 (2) = 6$	$a_{23} = 2 + 2 (3) = 8$

So the matrix is:

$A = [345678]$

Common Formula Types in FBISE Exercises

Formula	Meaning
$a_{ij} = i + j$	Sum of row and column indices
$a_{ij} = i \cdot j$	Product of indices
$a_{ij} = i^{2} - j$	Row index squared minus column index
$a_{ij} = \frac{i - j}{2}$	Difference divided by 2
$a_{ij} = {10 i = j i \neq = j$	Identity matrix pattern

Important Notes

Always identify the order of the matrix first to know the range of $i$ and $j$ .
Substitute carefully — do not swap $i$ and $j$ .
The resulting matrix may have real or complex entries depending on the formula.

Exercise 2.2 — Question 3

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.

Key Convention:

$i$ = row number, where $1 \leq i \leq m$
$j$ = column number, where $1 \leq j \leq n$
The element $a_{ij}$ occupies row $i$ and column $j$ .

Typical Problem Format

Construct a matrix of order $2 \times 3$ if $a_{ij}$ is defined by a given rule, for example:

$a_{ij} = i + 2 j$

Step-by-step solution:

For a $2 \times 3$ matrix, $i \in {1, 2}$ and $j \in {1, 2, 3}$ .

Compute each element:

	$j = 1$	$j = 2$	$j = 3$
$i = 1$	$a_{11} = 1 + 2 (1) = 3$	$a_{12} = 1 + 2 (2) = 5$	$a_{13} = 1 + 2 (3) = 7$
$i = 2$	$a_{21} = 2 + 2 (1) = 4$	$a_{22} = 2 + 2 (2) = 6$	$a_{23} = 2 + 2 (3) = 8$

So the matrix is:

$A = [345678]$

Common Formula Types in FBISE Exercises

Formula	Meaning
$a_{ij} = i + j$	Sum of row and column indices
$a_{ij} = i \cdot j$	Product of indices
$a_{ij} = i^{2} - j$	Row index squared minus column index
$a_{ij} = \frac{i - j}{2}$	Difference divided by 2
$a_{ij} = {10 i = j i \neq = j$	Identity matrix pattern

Important Notes

Always identify the order of the matrix first to know the range of $i$ and $j$ .
Substitute carefully — do not swap $i$ and $j$ .
The resulting matrix may have real or complex entries depending on the formula.