Exercise 3.2 · Q20: Bookshop stock value

Basics you need

Solution

1. Convert dozens to books

   $$\begin{aligned} \text{Chemistry}&: 10 \times 12 = 120\\ \text{Physics}&: 8 \times 12 = 96\\ \text{Economics}&: 10 \times 12 = 120 \end{aligned}$$
   The prices are "each", that is, per book. Multiplying a number of dozens by a price per book would give an answer 12 times too small. So first count books.

2. Build the two matrices

   Keep the order chemistry, physics, economics throughout.

   	Chemistry	Physics	Economics
   Number of books	120	96	120
   Price per book (₹)	80	60	40

   Quantity matrix $X$ (one row; columns: chemistry, physics, economics) and price matrix $P$ (one column; rows: chemistry, physics, economics):

   $$X = \left[\;120 \quad 96 \quad 120\;\right], \quad P = \begin{bmatrix} 80 \\ 60 \\ 40 \end{bmatrix}$$
   The $j$-th column of $X$ and the $j$-th row of $P$ both refer to the same subject. That is what makes the row-times-column product pair each quantity with its own price.
   Row matrix $1\times n$⁠, column matrix $n\times1$⁠. Lesson: types of matrices

3. Check the orders and choose the product order

   $X$ is $1\times3$ and $P$ is $3\times1$⁠. The columns of $X$ (3) equal the rows of $P$ (3), so $XP$ is defined and is $1\times1$⁠.

   We want one number, the total. $XP$ gives exactly one entry, the sum of the three "quantity × price" products. The other order, $PX$⁠, is $3\times3$ and contains meaningless entries such as $80 \times 96$ (chemistry price times number of physics books).
   $(1\times3)(3\times1) = 1\times1$⁠. Lesson: matrix multiplication

4. Multiply

   $$XP = \left[\;120 \quad 96 \quad 120\;\right]\begin{bmatrix} 80 \\ 60 \\ 40 \end{bmatrix}$$

   The single entry is row 1 of $X$ times column 1 of $P$⁠:

   $$\begin{aligned} c_{11} &= 120\cdot80 + 96\cdot60 + 120\cdot40\\ &= 9600 + 5760 + 4800\\ &= 20160 \end{aligned}$$

   So $XP = [20160]$⁠.

   Each product is "books × rupees per book = rupees": $120 \times 80 = 9600$ rupees from chemistry, $96 \times 60 = 5760$ from physics, $120 \times 40 = 4800$ from economics. Adding them gives the total in rupees.
   Entry of a row-times-column product: multiply matching entries and add. Basics: pair up, multiply, add

5. Interpret, and check by a second route

   In words: selling all the books brings in ₹20160.

   Second route, keeping dozens: the quantity matrix in books is 12 times the matrix in dozens, so

   $$XP = 12\left[\;10 \quad 8 \quad 10\;\right]\begin{bmatrix} 80 \\ 60 \\ 40 \end{bmatrix}$$ $$\begin{aligned} &= 12\,[10\cdot80 + 8\cdot60 + 10\cdot40]\\ &= 12\,[800 + 480 + 400]\\ &= 12\,[1680] = [20160] \end{aligned}$$

   Both routes agree.

   Write $D$ for the row of dozens, with entries 10, 8, 10, so that $X = 12D$⁠. A scalar can be taken outside a product: $(kD)P = k(DP)$⁠, because every term $k d_j p_j$ of the row-times-column sum carries the same factor $k$⁠. The second route reaches the same answer, which confirms the dozen conversion. (₹1680 is what the shop would receive if it had only 10, 8 and 10 single books.)

Every fact used, and where it comes from