Question

If matrix A = \[{{\text{[}}{{\text{a}}_{{\text{ij}}}}{\text{]}}_{{\ {3 \times 2}}}}\]​, and ${{\text{a}}_{{\text{ij}}}}$=\[{{\text{(3i - 2j)}}^2}\], then find the matrix A.

Answer 1

Hint – In order to solve this problem we need to get the value of \[{{\text{[}}{{\text{a}}_{{\text{ij}}}}{\text{]}}_{{\ {3 \times 2}}}}\]. To get the values we need to put the values of i and j in the equation given. As we know that this matrix has 3 rows and 2 columns the values of i will be 1, 2, and 3 and that of j will be 1 and 2.

Complete step by step answer:
The given equation is ${{\text{a}}_{{\text{ij}}}}$=\[{{\text{(3i - 2j)}}^2}\]
We need to find the values of A.
Where A = \[{{\text{[}}{{\text{a}}_{{\text{ij}}}}{\text{]}}_{{\ {3 \times 2}}}}\]
The total number of elements in A is 6.
The values of i will be 1, 2, and 3 and that of j will be 1 and 2.
The elements of the matrix are ${{\text{a}}_{11}},$${{\text{a}}_{{\text{12}}}}{\text{,}}\,\,{{\text{a}}_{{\text{21}}}}{\text{, }}{{\text{a}}_{{\text{22}}}}{\text{, }}{{\text{a}}_{{\text{31}}}}{\text{, }}{{\text{a}}_{{\text{32}}}}$.
The equation we have as ${{\text{a}}_{{\text{ij}}}}$=\[{{\text{(3i - 2j)}}^2}\]
$
  {a_{11}} = {(3(1) - 2(1))^2} = 1 \\
  {a_{12}} = {(3(1) - 2(2))^2} = 1 \\
  {a_{21}} = {(3(2) - 2(1))^2} = 16 \\
  {a_{22}} = {(3(2) - 2(2))^2} = 4 \\
  {a_{31}} = {(3(3) - 2(1))^2} = 49 \\
  {a_{32}} = {(3(3) - 2(2))^2} = 25 \\
$
So we found all the 6 elements of the matrix.
The matrix is : $\left[ \begin{gathered}
  {a_{11}}\,{a_{12}} \\
  {a_{21}}\,{a_{22}} \\
  {a_{31}}\,{a_{32}} \\
\end{gathered} \right]$= A
On putting the values calculated above we get the matrix as :
A = $\left[ \begin{gathered}
  1\,\,\,\,\,\,\,\,\,\,1 \\
  16\,\,\,\,\,\,4 \\
  49\,\,\,\,25 \\
\end{gathered} \right]$.

Note – Whenever you face such types of problems you should know that i is the number of rows whereas j is the number of columns and aij means the element of ith row and jth column. Then we used the equation given to get the elements of the matrix. Doing this you will get the right matrix.