Question

Let $f$ is a continuous function defined as $f:R\to Z,f\left(1\right)=2$ and a matrix $A=[a_{ij}{]}_{2\times 2}$ is defined as $a_{ij}=f\left(2i\right)+f\left(2j\right).$ Then the matrix $A$ is

• A. $\left[\begin{array}{cc}1& 1\\ 1& 1\end{array}\right]$
• B. $\left[\begin{array}{cc}2& 2\\ 2& 2\end{array}\right]$
• C. $\left[\begin{array}{cc}4& 4\\ 4& 4\end{array}\right]$
• D. $\left[\begin{array}{cc}8& 8\\ 8& 8\end{array}\right]$

Answer 1

The correct option is C $\left[\begin{array}{cc}4& 4\\ 4& 4\end{array}\right]$

As, $f$ is continuous in $R$ and $f\left(1\right)=2$ and range $\in Z$
$\Rightarrow f\left(x\right)=\text{constant}=2$
($\because$ function will become discontinuous if it takes other integral values)
$\Rightarrow a_{ij}=f\left(2i\right)+f\left(2j\right)$
$\Rightarrow a_{11}=4,a_{12}=4,a_{21}=4,a_{22}=4$
$\therefore A=\left[\begin{array}{cc}4& 4\\ 4& 4\end{array}\right]$