Question

Construct a \[2 \times 2\] matrix \[A\] \[ = \left[ {{a_{ij}}} \right]\] such that \[{a_{ij}} = i + 2j\].

Answer 1

Hint: A \[2 \times 2\]matrix is a matrix containing two rows and two columns. Now using the given relation, i.e. \[{a_{ij}} = i + 2j\], substitute the values of \[i,j\] and find the elements . Finally arrange the elements in the form of a matrix.

Complete step by step solution:
A matrix is an arrangement of variables or constants in rows and columns enclosed in square brackets. The order of a matrix denotes the number of rows and columns in it. The order of any matrix is denoted by \[m \times n\] where \[m\]denotes the number of rows and \[n\] denotes the number of columns. The position of elements of any matrix are generally denoted by \[{a_{ij}}\], where \[i\] represents the row number and \[j\] the column number.
The matrix to be constructed is a \[2 \times 2\] matrix, so it has two rows and two columns.
Now \[{a_{11}}\] is the element of the first row and first column.
\[{a_{12}}\] is the element of the first row and second column.
\[{a_{21}}\] is the element of the second row and first column.
\[{a_{22}}\] is the element of the second row and second column.
According to the given relation \[{a_{ij}} = i + 2j\]:
\[{a_{11}}\] \[ = \] \[1 + 2 = 3\]
\[{a_{12}} = 1 + 4 = 5\]
\[{a_{21}} = 2 + 2 = 4\]
\[{a_{22}} = 2 + 4 = 6\]
Hence arranging this elements in the form of a \[2 \times 2\] matrix:
\[A = \left[ {\begin{array}{*{20}{c}}
  {{a_{11}}}&{{a_{12}}} \\
  {{a_{21}}}&{{a_{22}}}
\end{array}} \right]\]

\[ \Rightarrow A = \left[ {\begin{array}{*{20}{c}}
  3&5 \\
  4&6
\end{array}} \right]\]

Note:
Students can make a mistake in understanding the position of the elements of the matrix. It must be carefully observed and always remember that \[i\] represents the row and \[j\] the column number respectively. Also note that for the order of the matrix if \[m = n\] then the matrix is called a square matrix.