Question

Consider two families $A$ and $B.$ Suppose there are $4$ men, $4$ women and $4$ children in family $A$ and $2$ men, $2$ women and $2$ children in family $B.$ The recommended daily amount of calories is $2400$ for a man, $1900$ for a woman, $1800$ for a child and $45$ grams of proteins for a man, $55$ grams for a woman and $33$ grams for a child.
The requirement of calories and proteins for each person is given by matrix $R$ and the number of family members in each family is given by matrix $F.$

If $F^T$ represents the transpose of matrix $F,$ then $R+100F^T$ is equal to

• A. $$\left[\begin{array}{cc}280& 245\\ 1900& 230\\ 2200& 233\end{array}\right]$$
• B. $$\left[\begin{array}{cc}2800& 255\\ 2400& 255\\ 2200& 233\end{array}\right]$$
• C. $$\left[\begin{array}{cc}2800& 45\\ 2300& 55\\ 2200& 33\end{array}\right]$$
• D. $$\left[\begin{array}{cc}2800& 245\\ 2300& 255\\ 2200& 233\end{array}\right]$$

Answer 1

The correct option is D $$\left[\begin{array}{cc}2800& 245\\ 2300& 255\\ 2200& 233\end{array}\right]$$

$R=\left[\begin{array}{cc}2400& 45\\ 1900& 55\\ 1800& 33\end{array}\right]$

$F=\left[\begin{array}{ccc}4& 4& 4\\ 2& 2& 2\end{array}\right]$

$\Rightarrow F^T=\left[\begin{array}{cc}4& 2\\ 4& 2\\ 4& 2\end{array}\right]$

$\Rightarrow 100F^T=100\left[\begin{array}{cc}4& 2\\ 4& 2\\ 4& 2\end{array}\right]=\left[\begin{array}{cc}400& 200\\ 400& 200\\ 400& 200\end{array}\right]$

Now, $R+100F^T$
$=\left[\begin{array}{cc}2400& 45\\ 1900& 55\\ 1800& 33\end{array}\right]+\left[\begin{array}{cc}400& 200\\ 400& 200\\ 400& 200\end{array}\right]=\left[\begin{array}{cc}2800& 245\\ 2300& 255\\ 2200& 233\end{array}\right]$