Question

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

Answer 1

Hint: First express the $2\times 2$ matrix in its standard form$\left|\begin{array}{cc}{a}_{11}& a_{12}\\ a_{21}& a_{22}\end{array}\right|$. Then calculate the elements of the matrix using the given formula $a_{ij}=\left|{\displaystyle \frac{3i-j}{2}}\right|$.

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 $\left|\begin{array}{cc}2& 5\\ 3& 6\end{array}\right|$.
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 $a_{ij}=\left|{\displaystyle \frac{3i-j}{2}}\right|$
We have to find a $2\times 2$ matrix.
So the matrix will be in the form $\left|\begin{array}{cc}{a}_{11}& a_{12}\\ a_{21}& a_{22}\end{array}\right|$
Now we have to calculate $a_{11,}{a}_{12},a_{21},a_{22}$ respectively.
$a_{ij}=\left|{\displaystyle \frac{3i-j}{2}}\right|$
So $a_{11}=\left|{\displaystyle \frac{3\times 1-1}{2}}\right|={\displaystyle \frac{2}{2}}=1$
Similarly $a_{12}=\left|{\displaystyle \frac{3\times 1-2}{2}}\right|={\displaystyle \frac{1}{2}}$, $a_{21}=\left|{\displaystyle \frac{3\times 2-1}{2}}\right|={\displaystyle \frac{5}{2}}$ and $a_{22}=\left|{\displaystyle \frac{3\times 2-2}{2}}\right|={\displaystyle \frac{4}{2}}=2$
So the required $2\times 2$ matrix is $\left|\begin{array}{cc}{a}_{11}& a_{12}\\ a_{21}& a_{22}\end{array}\right|$
=$\left|\begin{array}{cc}{a}_{11}& a_{12}\\ a_{21}& a_{22}\end{array}\right|=\left|\begin{array}{cc}1& {\displaystyle \frac{1}{2}}\\ {\displaystyle \frac{5}{2}}& 2\end{array}\right|$

The required $2\times 2$ matrix is $\left|\begin{array}{cc}1& {\displaystyle \frac{1}{2}}\\ {\displaystyle \frac{5}{2}}& 2\end{array}\right|$

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 $a_{ij}=0$.