Answer to Question #168897 in Linear Algebra for JoshColl

Question

Mr. Akpalu went to a market to purchase 3 kg of sugar, 10 kg of wheat and 1 kg of salt. In a shop near to Mr. Akpalu’s residence, these commodities are priced at € 20, €10 and € 6 per kg whereas in the local market these commodities are priced at € 15, € 8 and € 6 per kg respectively. If the cost of traveling to local market is € 25, find the net savings of Mr. Akpalu, using matrix multiplication method.

Answer 1

Denote by $C_1$ and $C_2$ vectors of prices in a shop and in the market respectively. Then

$$C_1=\begin{pmatrix} 20 \\ 10 \\ 6 \end{pmatrix},C_2=\begin{pmatrix} 15 \\ 8 \\ 6 \end{pmatrix}$$

Let $Q$ denotes vector of amounts needed for Mr.Akpalu. Then

$$Q=\begin{pmatrix} 3 \\ 10 \\ 1 \end{pmatrix}$$

Then the total price he pays equals to $Q(C_i)$

Thus the difference between total prices in 2 shops equals to

$$Q(C_1)-Q(C_2)=Q(C_1-C_2)=$$

$$\begin{pmatrix} 3 & 10 & 1 \end{pmatrix}\bigg(\begin{pmatrix} 20 \\ 10 \\ 6 \end{pmatrix}-\begin{pmatrix} 15 \\ 8 \\ 6 \end{pmatrix}\bigg)$$

$$\begin{pmatrix} 3 & 10 & 1 \end{pmatrix}\begin{pmatrix} 5 \\ 2 \\ 0 \end{pmatrix}=3(5)+10(2)+1(0)=35$$

Since travelling to local market costs € 25 net savings equals to

$$€ 35-€ 25=€ 10$$

The net saving of Mr.Akpalu is € 10.