2.2 Q-5 | Matrices And Determinants | Mathematics 11

Exercise 2.2 — Question 5

Construct a matrix $A = [a_{ij}]$ of order $3 \times 3$ where the elements are defined by:

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

Method

For a $3 \times 3$ matrix, the row index $i$ runs from $1$ to $3$ and the column index $j$ runs from $1$ to $3$ . Substitute each pair $(i, j)$ into the formula to find every element.

$A = a_{11} a_{21} a_{31} a_{12} a_{22} a_{32} a_{13} a_{23} a_{33}$

Calculating Each Element

Element	$i$	$j$	Calculation	Value
$a_{11}$	1	1	$\frac{( 1 + 1 ) ^{2}}{2} = \frac{4}{2}$	$2$
$a_{12}$	1	2	$\frac{( 1 + 2 ) ^{2}}{2} = \frac{9}{2}$	$\frac{9}{2}$
$a_{13}$	1	3	$\frac{( 1 + 3 ) ^{2}}{2} = \frac{16}{2}$	$8$
$a_{21}$	2	1	$\frac{( 2 + 1 ) ^{2}}{2} = \frac{9}{2}$	$\frac{9}{2}$
$a_{22}$	2	2	$\frac{( 2 + 2 ) ^{2}}{2} = \frac{16}{2}$	$8$
$a_{23}$	2	3	$\frac{( 2 + 3 ) ^{2}}{2} = \frac{25}{2}$	$\frac{25}{2}$
$a_{31}$	3	1	$\frac{( 3 + 1 ) ^{2}}{2} = \frac{16}{2}$	$8$
$a_{32}$	3	2	$\frac{( 3 + 2 ) ^{2}}{2} = \frac{25}{2}$	$\frac{25}{2}$
$a_{33}$	3	3	$\frac{( 3 + 3 ) ^{2}}{2} = \frac{36}{2}$	$18$

Result

$A = 2 \frac{9}{2} 8 \frac{9}{2} 8 \frac{25}{2} 8 \frac{25}{2} 18$

Key Observation: Notice that $a_{ij} = a_{j i}$ for all $i, j$ — this matrix is symmetric because the formula $a_{ij} = \frac{( i + j ) ^{2}}{2}$ is symmetric in $i$ and $j$ .

Exercise 2.2 — Question 5

Construct a matrix $A = [a_{ij}]$ of order $3 \times 3$ where the elements are defined by:

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

Method

For a $3 \times 3$ matrix, the row index $i$ runs from $1$ to $3$ and the column index $j$ runs from $1$ to $3$ . Substitute each pair $(i, j)$ into the formula to find every element.

$A = a_{11} a_{21} a_{31} a_{12} a_{22} a_{32} a_{13} a_{23} a_{33}$

Calculating Each Element

Element	$i$	$j$	Calculation	Value
$a_{11}$	1	1	$\frac{( 1 + 1 ) ^{2}}{2} = \frac{4}{2}$	$2$
$a_{12}$	1	2	$\frac{( 1 + 2 ) ^{2}}{2} = \frac{9}{2}$	$\frac{9}{2}$
$a_{13}$	1	3	$\frac{( 1 + 3 ) ^{2}}{2} = \frac{16}{2}$	$8$
$a_{21}$	2	1	$\frac{( 2 + 1 ) ^{2}}{2} = \frac{9}{2}$	$\frac{9}{2}$
$a_{22}$	2	2	$\frac{( 2 + 2 ) ^{2}}{2} = \frac{16}{2}$	$8$
$a_{23}$	2	3	$\frac{( 2 + 3 ) ^{2}}{2} = \frac{25}{2}$	$\frac{25}{2}$
$a_{31}$	3	1	$\frac{( 3 + 1 ) ^{2}}{2} = \frac{16}{2}$	$8$
$a_{32}$	3	2	$\frac{( 3 + 2 ) ^{2}}{2} = \frac{25}{2}$	$\frac{25}{2}$
$a_{33}$	3	3	$\frac{( 3 + 3 ) ^{2}}{2} = \frac{36}{2}$	$18$

Result

$A = 2 \frac{9}{2} 8 \frac{9}{2} 8 \frac{25}{2} 8 \frac{25}{2} 18$

Key Observation: Notice that $a_{ij} = a_{j i}$ for all $i, j$ — this matrix is symmetric because the formula $a_{ij} = \frac{( i + j ) ^{2}}{2}$ is symmetric in $i$ and $j$ .