Answer:
<h2>See the explanation.</h2>
Step-by-step explanation:
(a)
![\left[\begin{array}{cccc}T&H&S&P\\S&600&1300&2000\\L&400&300&400\end{array}\right]](https://tex.z-dn.net/?f=%5Cleft%5B%5Cbegin%7Barray%7D%7Bcccc%7DT%26H%26S%26P%5C%5CS%26600%261300%262000%5C%5CL%26400%26300%26400%5Cend%7Barray%7D%5Cright%5D)
= A.
In the above matrix A, the columns refers the three type of books and the rows refers the from which stores the books are been sold.
The numbers represents the corresponding sales in the month of January.
The sale is same for the 6 months.
Hence, 6A = ![\left[\begin{array}{cccc}T&H&S&P\\S&3600&7800&12000\\L&2400&1800&2400\end{array}\right]](https://tex.z-dn.net/?f=%5Cleft%5B%5Cbegin%7Barray%7D%7Bcccc%7DT%26H%26S%26P%5C%5CS%263600%267800%2612000%5C%5CL%262400%261800%262400%5Cend%7Barray%7D%5Cright%5D)
. This matrix 6A represents the total sales over the 6 months.
(b)
If we denote the books in stock at the starting of January by B, then
B = ![\left[\begin{array}{cccc}T&H&S&P\\S&1000&3000&6000\\L&1000&6000&3000\end{array}\right]](https://tex.z-dn.net/?f=%5Cleft%5B%5Cbegin%7Barray%7D%7Bcccc%7DT%26H%26S%26P%5C%5CS%261000%263000%266000%5C%5CL%261000%266000%263000%5Cend%7Barray%7D%5Cright%5D)
.
Each month, the chain restocked the stores from its warehouse by shipping 500 hardcover, 1,400 softcover, and 1,400 plastic books to San Francisco and 500 hardcover, 500 softcover, and 500 plastic books to Los Angeles.
If we represent the amount restocked books at the end of each month by another matrix C, then
C = ![\left[\begin{array}{cccc}T&H&S&P\\S&500&1400&1400\\L&500&500&500\end{array}\right]](https://tex.z-dn.net/?f=%5Cleft%5B%5Cbegin%7Barray%7D%7Bcccc%7DT%26H%26S%26P%5C%5CS%26500%261400%261400%5C%5CL%26500%26500%26500%5Cend%7Barray%7D%5Cright%5D)
.
This restocking will be done for 5 times before the end of June.
If there would be no sale, then the stock would be
B + 5C = ![\left[\begin{array}{cccc}T&H&S&P\\S&1000+2500&3000+7000&6000+7000\\L&1000+2500&6000+2500&3000+2500\end{array}\right] \\= \left[\begin{array}{cccc}T&H&S&P\\S&3500&10000&13000\\L&3500&8500&5500\end{array}\right]](https://tex.z-dn.net/?f=%5Cleft%5B%5Cbegin%7Barray%7D%7Bcccc%7DT%26H%26S%26P%5C%5CS%261000%2B2500%263000%2B7000%266000%2B7000%5C%5CL%261000%2B2500%266000%2B2500%263000%2B2500%5Cend%7Barray%7D%5Cright%5D%20%5C%5C%3D%20%5Cleft%5B%5Cbegin%7Barray%7D%7Bcccc%7DT%26H%26S%26P%5C%5CS%263500%2610000%2613000%5C%5CL%263500%268500%265500%5Cend%7Barray%7D%5Cright%5D)
.
Since, the total sale is given by 6A, at the end of June, the inventory in each store can be shown as following,
B+5C-6A ![\left[\begin{array}{cccc}T&H&S&P\\S&3500&10000&13000\\L&3500&8500&5500\end{array}\right] - \left[\begin{array}{cccc}T&H&S&P\\S&3600&7800&12000\\L&2400&1800&2400\end{array}\right] \\= \left[\begin{array}{cccc}T&H&S&P\\S&-100&2200&1000\\L&1100&6700&3100\end{array}\right]](https://tex.z-dn.net/?f=%5Cleft%5B%5Cbegin%7Barray%7D%7Bcccc%7DT%26H%26S%26P%5C%5CS%263500%2610000%2613000%5C%5CL%263500%268500%265500%5Cend%7Barray%7D%5Cright%5D%20-%20%5Cleft%5B%5Cbegin%7Barray%7D%7Bcccc%7DT%26H%26S%26P%5C%5CS%263600%267800%2612000%5C%5CL%262400%261800%262400%5Cend%7Barray%7D%5Cright%5D%20%5C%5C%3D%20%5Cleft%5B%5Cbegin%7Barray%7D%7Bcccc%7DT%26H%26S%26P%5C%5CS%26-100%262200%261000%5C%5CL%261100%266700%263100%5Cend%7Barray%7D%5Cright%5D)
- translation a units to the right;
- translation a units to the left;
- translation a units up;
- translation a units down.
