Question

Laura is planning to buy two 5-lb bags of sugar, three 5-b bags of flour, two 1-gal cartons of milk, and three 1-dozen cartons of large eggs. The prices of these items in three neighborhood supermarkets are as follows Milk Eggs (1-doren carton) Sugar (5-lb bag) Flour (1-gal carton) (5-lb bag) Supermarket I Supermarket II Supermarket 11 $3.15 $3.79 $2.99 $3.49 $2.99 $2.89 $2.79 $3.29 $3.74 $2.98 $2.89 $2.99 (a) Write a 3x 4 matrix A to represent the prices (in dollars) of the items in the three supermarkets Am (b) Write a 4x1 matrix B to represent the quantities of sugar, flour, milk, and eggs that Laura plans to purchase in the three supermarketts (c) Use matrix multiplication to find a matrix C that represents Laura's total outlay (in dollars) at each supermarket C-
At which supermarket should she make her purchase if she wants to minimize her cost? (Assume that she will shop at only one supermarket.) O supermarket I O supermarket II O supermarket III Need Help? Read It

Answer 1

Answer:

(a, b) see the input matrices in the calculator image below

(c) see the output matrix in the calculator image below

Laura should use Supermarket II.

Step-by-step explanation:

(a) Your data is not clearly identified, so we have assumed that the item costs are listed for one supermarket before they are listed for the next. Then your matrix A will be ...

$A=\left[\begin{array}{cccc}3.15&3.79&2.99&3.49\\2.99&2.89&2.79&3.29\\3.74&2.98&2.89&2.99\end{array}\right]$

__

(b) The column matrix of purchase amounts will be ...

$B=\left[\begin{array}{c}2&3&2&3\end{array}\right]$

__

(c) It is somewhat tedious to do matrix multiplication by hand, so we have let a calculator do it. Some calculators offer easier data entry than others, and some insist that data be entered into tables before any calculation can be done. We have chosen this one (attached), not because its use is easiest, but because we can post a picture of the entry and the result.

$C=\left[\begin{array}{c}34.12&30.10&31.17\end{array}\right]$

Laura's total bill is least at Supermarket II.

_____

As you know, the row-column result of matrix multiplication is the element-by-element product of the 'row' of the left matrix by the 'column' of the right matrix. Here, that means the 2nd row 1st column of the output is computed from the 2nd row of A and the 1st (only) column of B:

  2.99·2 +2.89·3 +2.79·2 +3.29·3 = 5.98 +8.67 +5.58 +9.87

  = 30.10