Chapter 3.5, Problem 21PPS

To determine

To express each library’s number of books as matrix, then find out total number of books of each type and also more books in Library A than in C.

Answer to Problem 21PPS

(a) $\text{A}=\left[\begin{array}{c}10000\\ 5000\\ 5000\end{array}\right]$ , $\text{B}=\left[\begin{array}{c}15000\\ 10000\\ 2500\end{array}\right]$ , $\text{C}=\left[\begin{array}{c}4000\\ 700\\ 800\end{array}\right]$

(b) Total number of, Novels = 29000

Biographies = 15700

Children’s books = 8300

(c) Library A has more books than C, by,

Novels = 6000

Biographies = 4300

Children’s books = 4200

Explanation of Solution

Given Information

Library A,	Novels = 10000
		Biographies = 5000
		Children’s books = 5000

Library B,	Novels = 15000
		Biographies = 10000
		Children’s books = 2500

Library C,	Novels = 4000
		Biographies = 700
		Children’s books = 800

Calculation

(a)

We write matrix A, B and C as per the respective names of the library.

So,

  $\text{A}=\left[\begin{array}{c}10000\\ 5000\\ 5000\end{array}\right]$

  $\text{B}=\left[\begin{array}{c}15000\\ 10000\\ 2500\end{array}\right]$

  $\text{C}=\left[\begin{array}{c}4000\\ 700\\ 800\end{array}\right]$

(b)

For finding out the total number of books of each type, we will have to add all the three matrices,

  So,  A + B + C = $\left[\begin{array}{c}10000\\ 5000\\ 5000\end{array}\right]+\left[\begin{array}{c}15000\\ 10000\\ 2500\end{array}\right]+\left[\begin{array}{c}4000\\ 700\\ 800\end{array}\right]$

  ⇒  A + B + C = $\left[\begin{array}{c}29000\\ 15700\\ 8300\end{array}\right]$

  Hence, Total of, Novels = 29000

    Biographies = 15700

   Children’s books = 8300

(c)

For the calculation of, more books in library A than in C, we will have to subtract both the matrices,

So,

A − C = $\left[\begin{array}{c}10000\\ 5000\\ 5000\end{array}\right]-\left[\begin{array}{c}4000\\ 700\\ 800\end{array}\right]$

  = $\left[\begin{array}{c}6000\\ 4300\\ 4200\end{array}\right]$

Therefore, Library A has more books than C, by,

  Novels = 6000

  Biographies = 4300

  Children’s books = 4200

(d)

A + B = $\left[\begin{array}{c}10000\\ 5000\\ 5000\end{array}\right]+\left[\begin{array}{c}15000\\ 10000\\ 2500\end{array}\right]$

  = $\left[\begin{array}{c}25000\\ 15000\\ 7500\end{array}\right]$

The matrix, A + B, shows the total number of books of each kind in both the libraries, as

In library A and B, we have total,

  Novels = 25000

  Biographies = 15000

  Children’s books = 7500