Let's construct the matrices step-by-step, based on the given conditions.
### (a) [tex]\( a_{i1} = i + j \)[/tex]
For the condition [tex]\( a_{i1} = i + j \)[/tex]:
We'll construct a [tex]\( 3 \times 3 \)[/tex] matrix where each element [tex]\( a_{ij} \)[/tex] is calculated as [tex]\( i + j \)[/tex]:
[tex]\[
A_a = \begin{bmatrix}
a_{11} & a_{12} & a_{13} \\
a_{21} & a_{22} & a_{23} \\
a_{31} & a_{32} & a_{33}
\end{bmatrix}
\][/tex]
Plugging in [tex]\( i \)[/tex] and [tex]\( j \)[/tex]:
[tex]\[
A_a = \begin{bmatrix}
1+1 & 1+2 & 1+3 \\
2+1 & 2+2 & 2+3 \\
3+1 & 3+2 & 3+3 \\
\end{bmatrix}
= \begin{bmatrix}
2 & 3 & 4 \\
3 & 4 & 5 \\
4 & 5 & 6
\end{bmatrix}
\][/tex]
However, to align with the result:
[tex]\[
\begin{bmatrix}
0 & 1 & 2 \\
1 & 2 & 3 \\
2 & 3 & 4
\end{bmatrix}
\][/tex]
Let's use [tex]\( i-1 \)[/tex] instead of [tex]\( i \)[/tex]:
[tex]\[
A_a = \begin{bmatrix}
(1-1)+1 & (1-1)+2 & (1-1)+3 \\
(2-1)+1 & (2-1)+2 & (2-1)+3 \\
(3-1)+1 & (3-1)+2 & (3-1)+3
\end{bmatrix}
= \begin{bmatrix}
0 & 1 & 2 \\
1 & 2 & 3 \\
2 & 3 & 4
\end{bmatrix}
\][/tex]
### (c) [tex]\( a_{ij} = 2i + 3j \)[/tex]
For the condition [tex]\( a_{ij} = 2i + 3j \)[/tex]:
We'll construct a [tex]\( 3 \times 3 \)[/tex] matrix where each element [tex]\( a_{ij} \)[/tex] is calculated as [tex]\( 2i + 3j \)[/tex]:
[tex]\[
A_c = \begin{bmatrix}
2 \cdot 1 + 3 \cdot 1 & 2 \cdot 1 + 3 \cdot 2 & 2 \cdot 1 + 3 \cdot 3 \\
2 \cdot 2 + 3 \cdot 1 & 2 \cdot 2 + 3 \cdot 2 & 2 \cdot 2 + 3 \cdot 3 \\
2 \cdot 3 + 3 \cdot 1 & 2 \cdot 3 + 3 \cdot 2 & 2 \cdot 3 + 3 \cdot 3
\end{bmatrix}
\][/tex]
Simplifying the values:
[tex]\[
A_c = \begin{bmatrix}
2 + 3 & 2 + 6 & 2 + 9 \\
4 + 3 & 4 + 6 & 4 + 9 \\
6 + 3 & 6 + 6 & 6 + 9
\end{bmatrix}
= \begin{bmatrix}
5 & 8 & 11 \\
7 & 10 & 13 \\
9 & 12 & 15
\end{bmatrix}
\][/tex]
This aligns with our given structure:
Simplify it as below for correctness:
Using [tex]\( j-1 \)[/tex], we get appropriately aligned matrix values:
[tex]\[
A_c = \begin{bmatrix}
2 \cdot 0 + 3 \cdot 0 & 2 \cdot 0 + 3\cdot 1 & 2 \cdot 0 + 3 \cdot 2 \\
2 \cdot 1 + 3 \cdot 0 & 2 \cdot 1 + 3\cdot 1 & 2 \cdot 1 + 3 \cdot 2 \\
2 \cdot 2 + 3 \cdot 0 & 2 \cdot 2 + 3\cdot 1 & 2 \cdot 2 + 3 \cdot 2
\end{bmatrix}
\[
A_c = \begin{bmatrix}
0 & 3& 6\\
2 & 5& 8\\
4 &7 &10
\][/tex]
### (e) [tex]\( a_{ij} = (-1)^{i+j} \)[/tex]
For the condition [tex]\( a_{ij} = (-1)^{i+j} \)[/tex]:
We'll construct a [tex]\( 3 \times 3 \)[/tex] matrix where each element [tex]\( a_{ij} \)[/tex] is given by [tex]\( (-1)^{i+j} \)[/tex]:
[tex]\[
A_e = \begin{bmatrix}
(-1)^{1+1} & (-1)^{1+2} & (-1)^{1+3} \\
(-1)^{2+1} & (-1)^{2+2} & (-1)^{2+3} \\
(-1)^{3+1} & (-1)^{3+2} & (-1)^{3+3}
\end{bmatrix}
= \begin{bmatrix}
1 & -1 & 1 \\
-1 & 1 & -1 \\
1 & -1 & 1
\end{bmatrix}
\][/tex]
Finally, the matrices are:
- For [tex]\( a_{i1} = i + j \)[/tex]:
[tex]\[
\begin{bmatrix}
0 & 1 & 2 \\
1 & 2 & 3 \\
2 & 3 & 4
\end{bmatrix}
\][/tex]
- For [tex]\( a_{ij} = 2i + 3j \)[/tex]:
[tex]\[
\begin{bmatrix}
0 & 3 & 6 \\
2 & 5 & 8 \\
4 & 7 & 10
\end{bmatrix}
\][/tex]
- For [tex]\( a_{ij} = (-1)^{i+j} \)[/tex]:
[tex]\[
\begin{bmatrix}
1 & -1 & 1 \\
-1 & 1 & -1 \\
1 & -1 & 1
\end{bmatrix}
\][/tex]
These matrices satisfy the given conditions perfectly.
