Question

Construct a 2 × 2 matrix whose elements are given by \(a_{ij} = \frac{(i + 2j)^2}{2}\)
1. \(\left[ {\begin{array}{*{20}{c}} {\frac{9}{2}}&{\frac{{25}}{2}}\\ 5&{18} \end{array}} \right]\)
2. \(\left[ {\begin{array}{*{20}{c}} {\frac{9}{2}}&{\frac{{25}}{2}}\\ 8&{9} \end{array}} \right]\)
3. \(\left[ {\begin{array}{*{20}{c}} {\frac{9}{2}}&{\frac{{25}}{2}}\\ 8&{18} \end{array}} \right]\)
4. \(\left[ {\begin{array}{*{20}{c}} {\frac{7}{2}}&{\frac{{25}}{2}}\\ 8&{18} \end{array}} \right]\)
5. None of these

Answer 1

Correct Answer - Option 3 : \(\left[ {\begin{array}{*{20}{c}} {\frac{9}{2}}&{\frac{{25}}{2}}\\ 8&{18} \end{array}} \right]\)

Concept:

A rectangular representation of mn numbers (complex or real) in the form of m rows and n columns is called a matrix of order m × n.

Any m × n matrix is represented as, \(A = \;{\left[ {\begin{array}{*{20}{c}} {{a_{11}}}& \cdots &{{a_{1n}}}\\ \vdots & \ddots & \vdots \\ {{a_{m1}}}& \cdots &{{a_{mn}}} \end{array}} \right]_{m × \;n}}\)

Calculation:

Let A be the matrix whose is 2 × 2 matrix and the elements of the matrix are given by \(a_{ij} = \frac{(i + 2j)^2}{2}\).

∵ The order of the matrix A is 2 × 2 i.e no. of rows = 2 and no. of columns = 2

So, 1 ≤ i ∈ N ≤ 2 and 1 ≤ j ∈ N ≤ 2.

∵ \(a_{ij} = \frac{(i + 2j)^2}{2}\)

⇒ a₁₁ = 9/2, a₁₂ = 25/2, a₂₁ = 8 and a₂₂ = 18

Hence, \(A = \left[ {\begin{array}{*{20}{c}} {\frac{9}{2}}&{\frac{{25}}{2}}\\ 8&{18} \end{array}} \right]\)