A certain bookstore chain has two stores, one in San Francisco and one in Los Angeles. It stocks three kinds of books: hardcover, softcover, and plastic (for infants). At the beginning of January, the central computer showed the following books in stock. Hard Soft Plastic
San Francisco 1,000 3,000 6,000
Los Angeles 1,000 6,000 3,000

Its sales in January were as follows: 700 hardcover books, 1,200 softcover books, and 2,000 plastic books sold in San Francisco, and 400 hardcover, 200 softcover, and 500 plastic books sold in Los Angeles. The bookstore chain actually maintained the same sales figures for the first 6 months of the year. Each month, the chain restocked the stores from its warehouse by shipping 600 hardcover, 1,500 softcover, and 1,500 plastic books to San Francisco and 500 hardcover, 500 softcover, and 500 plastic books to Los Angeles.

Required:
a. Use matrix operations to determine the total sales over the 6 months, broken down by store and type of book.
b. Use matrix operations to determine the inventory in each store at the end of June.

a. Sales from January to June: Matrix B6

                             Hard    Soft       Plastic

San Francisco     4,200   7,200     12,000

Los Angeles       2,400    1,200      3,000

b) Ending Inventory: Matrix D:

                           Hard      Soft      Plastic

San Francisco      400     4,800     3,000

Los Angeles      1,600     7,800     3,000

Step-by-step explanation:

a) Data and Calculations:

Stock on January 1: Matrix A

                              Hard    Soft       Plastic

San Francisco       1,000   3,000    6,000

Los Angeles         1,000   6,000    3,000

Sales in January: Matrix B

                             Hard    Soft       Plastic

San Francisco       700     1,200     2,000

Los Angeles         400       200        500

Restocking: Matrix C

                          Hard    Soft       Plastic

San Francisco    600   1,500      1,500

Los Angeles      500     500        500

Total Sales over the first 6 months =

Matrix B * 6 = Matrix B6

Sales in January: Matrix B

                             Hard    Soft       Plastic

San Francisco       700     1,200     2,000

Los Angeles         400       200        500

* 6

=

Sales from January to June: Matrix B6

                            Hard    Soft       Plastic

San Francisco     4,200   7,200     12,000

Los Angeles       2,400    1,200      3,000

Matrix C6 = Matrix C * 6

=

Restocking: Matrix C6

                          Hard     Soft       Plastic

San Francisco  3,600   9,000      9,000

Los Angeles    3,000   3,000      3,000

Inventory at the end of June =

Matrix A + Matrix C6 - Matrix B6

= Matrix D

Stock on January 1: Matrix A

                             Hard    Soft       Plastic

San Francisco       1,000   3,000    6,000

Los Angeles         1,000   6,000    3,000

+

Restocking: Matrix C6

                           Hard     Soft       Plastic

San Francisco  3,600   9,000      9,000

Los Angeles    3,000   3,000      3,000

-

Sales from January to June: Matrix B6

                             Hard    Soft       Plastic

San Francisco     4,200   7,200     12,000

Los Angeles       2,400    1,200      3,000

Ending Inventory: Matrix D:

                            Hard      Soft       Plastic

San Francisco      400     4,800     3,000

Los Angeles      1,600     7,800     3,000

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