To construct the matrix [tex]\(A\)[/tex] defined as [tex]\(A = (a_{ij})_{2 \times 3}\)[/tex] with the formula for the elements [tex]\(a_{ij} = (i \times j)^2\)[/tex], follow these steps:
1. Determine the dimensions of the matrix:
Matrix [tex]\(A\)[/tex] is a [tex]\(2 \times 3\)[/tex] matrix, meaning it has 2 rows and 3 columns.
2. Identify the formula:
The elements of the matrix are given by the formula [tex]\(a_{ij} = (i \times j)^2\)[/tex], where [tex]\(i\)[/tex] represents the row number and [tex]\(j\)[/tex] represents the column number.
3. Calculate each element of the matrix:
- For [tex]\(i = 1\)[/tex] and [tex]\(j = 1\)[/tex], [tex]\(a_{11} = (1 \times 1)^2 = 1^2 = 1\)[/tex]
- For [tex]\(i = 1\)[/tex] and [tex]\(j = 2\)[/tex], [tex]\(a_{12} = (1 \times 2)^2 = 2^2 = 4\)[/tex]
- For [tex]\(i = 1\)[/tex] and [tex]\(j = 3\)[/tex], [tex]\(a_{13} = (1 \times 3)^2 = 3^2 = 9\)[/tex]
- For [tex]\(i = 2\)[/tex] and [tex]\(j = 1\)[/tex], [tex]\(a_{21} = (2 \times 1)^2 = 2^2 = 4\)[/tex]
- For [tex]\(i = 2\)[/tex] and [tex]\(j = 2\)[/tex], [tex]\(a_{22} = (2 \times 2)^2 = 4^2 = 16\)[/tex]
- For [tex]\(i = 2\)[/tex] and [tex]\(j = 3\)[/tex], [tex]\(a_{23} = (2 \times 3)^2 = 6^2 = 36\)[/tex]
4. Construct the matrix:
Now, place the calculated elements into the matrix according to their positions:
[tex]\[
A = \begin{pmatrix}
a_{11} & a_{12} & a_{13} \\
a_{21} & a_{22} & a_{23} \\
\end{pmatrix}
=
\begin{pmatrix}
1 & 4 & 9 \\
4 & 16 & 36 \\
\end{pmatrix}
\][/tex]
Therefore, the constructed matrix [tex]\(A\)[/tex] is:
[tex]\[
A = \begin{pmatrix}
1 & 4 & 9 \\
4 & 16 & 36 \\
\end{pmatrix}
\][/tex]
