Question

Elements of a matrix is represented by ${a_{ij}} = \left| {\dfrac{{3i - j}}{2}} \right|$. Find the $2 \times 2$ matrix.

Answer 1

First express the $2 \times 2$ matrix in its standard form\left| {\begin{array}{*{20}{c}} {{a_{11}}}&{{a_{12}}} \\\ {{a_{21}}}&{{a_{22}}} \end{array}} \right|. Then calculate the elements of the matrix using the given formula ${a_{ij}} = \left| {\dfrac{{3i - j}}{2}} \right|$.

Complete step-by-step answer:
In mathematics, a matrix (plural matrices) is a rectangular array or table of numbers, symbols, or expressions, arranged in rows and columns. Example of a $2 \times 2$matrix is \left| {\begin{array}{*{20}{c}} 2&5 \\\ 3&6 \end{array}} \right|.
The size of a matrix is denoted by the number of rows and columns that a matrix contains.
The elements of a matrix is denoted by ${a_{ij}}$, it means that ${a_{ij}}$is the element in the $i$th row and $j$th column.
Here it is mentioned that ${a_{ij}} = \left| {\dfrac{{3i - j}}{2}} \right|$
We have to find a $2 \times 2$ matrix.
So the matrix will be in the form \left| {\begin{array}{*{20}{c}} {{a_{11}}}&{{a_{12}}} \\\ {{a_{21}}}&{{a_{22}}} \end{array}} \right|
Now we have to calculate ${a_{11,}}{a_{12}},{a_{21}},{a_{22}}$ respectively.
${a_{ij}} = \left| {\dfrac{{3i - j}}{2}} \right|$
So ${a_{11}} = \left| {\dfrac{{3 \times 1 - 1}}{2}} \right| = \dfrac{2}{2} = 1$
Similarly ${a_{12}} = \left| {\dfrac{{3 \times 1 - 2}}{2}} \right| = \dfrac{1}{2}$, ${a_{21}} = \left| {\dfrac{{3 \times 2 - 1}}{2}} \right| = \dfrac{5}{2}$ and ${a_{22}} = \left| {\dfrac{{3 \times 2 - 2}}{2}} \right| = \dfrac{4}{2} = 2$
So the required $2 \times 2$ matrix is \left| {\begin{array}{*{20}{c}} {{a_{11}}}&{{a_{12}}} \\\ {{a_{21}}}&{{a_{22}}} \end{array}} \right|
=\left| {\begin{array}{*{20}{c}} {{a_{11}}}&{{a_{12}}} \\\ {{a_{21}}}&{{a_{22}}} \end{array}} \right| = \left| {\begin{array}{*{20}{c}} 1&{\dfrac{1}{2}} \\\ {\dfrac{5}{2}}&2 \end{array}} \right|

The required $2 \times 2$ matrix is \left| {\begin{array}{*{20}{c}} 1&{\dfrac{1}{2}} \\\ {\dfrac{5}{2}}&2 \end{array}} \right|

Note: We should also remember various information of matrix regarding ${a_{ij}}$
For example: If a matrix is symmetric than we can say that ${a_{ij}} = {a_{ji}}$, whether if the matrix be skew symmetric then ${a_{ij}} = - {a_{ji}}$. For a null matrix ${a_{ij}} = 0$.