Question

Write the element ${a_{2\,3}}$ of a matrix (${a_{i\,j}}$) whose elements are represented by ${a_{i\,j}} = \dfrac{{|i - j|}}{2}$

Answer 1

In matrix each element is represented by ${a_{i\,j}}$ , where i denotes the row and the j denotes the column in which the element is present. So, if an element is present in the first row and second column. Then the element is written as ${a_{12}}$. Since it is matrix denoted A. we can find the element in ${a_{2\,3}}$ by putting the values of i and j in ${a_{i\,j}} = \dfrac{{|i - j|}}{2}$

Stepwise solution:
Given:
Element ${a_{2\,3}}$ is present in a matrix elements are represented by ${a_{i\,j}} = \dfrac{{|i - j|}}{2}$
Steps:
A matrix is an ordered rectangular array of numbers or fractions. The numbers or functions are called the elements or entries of the matrix. A matrix having 3 rows and 3 columns is called a matrix of order . The position of an element is denoted by i and j where i denotes the row and j the column where the element is located. So, for ${a_{2\,3}}$ the value of i is 2 and j is 3. So, to find the element at position ${a_{2\,3}}$ we place the value of i and j in ${a_{ij}} = \dfrac{{|i - j|}}{2}$
$\Rightarrow {a_{23}} = \dfrac{{|2 - 3|}}{2} = \dfrac{1}{2}$
|i-j| is (i-j) inside a || (modulus), so only magnitude is considered while direction is not taken into consideration, that is negative sign is ignored.
Thus, the element at ${a_{2\,3}}$ in the order matrix A is $\dfrac{1}{2}$.

Note:
When the value of m and n is equal then we term the matrix a square matrix (m=n). A matrix is a very useful mathematical tool that has wide applications, from coding to quantum mechanics. Matrix addition, subtraction, multiplication is easy to learn. Students should go through them for better mathematical knowledge.