Question

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

Answer 1

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_{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.