Book edition 5th

Author(s) Otto Bretscher

Pages 442 pages

ISBN 9780321796974

Answers without the blur.

Just sign up for free and you're in.

Short Answer

Some parking meters in downtown Geneva, Switzerland, accept $2$ Franc and $5$ Franc coins.
a. A parking officer collects $51$ coins worth $144$ Francs. How many coins are there of each kind?
b. Find the matrix $A$ that transforms the vector
$[\begin{matrix} number of 2 Franc coins \\ number of 5 Franc coins \end{matrix}]$
into the vector
$[\begin{matrix} total value of coins \\ total number of coins \end{matrix}]$
c. Is the matrix $A$ in part (b) invertible? If so, find the inverse (use Exercise 13). Use the result to check your answer in part (a).

a. There are $37$ coins of $S F r 2$ and $14$ coins of $S F r 5$ .

b. The matrix $A$ is $A = [\begin{matrix} 2 & 5 \\ 1 & 1 \end{matrix}]$ .

c. The matrix $A$ is invertible and the inverse is $A^{- 1} = [\begin{matrix} - \frac{1}{3} & \frac{5}{3} \\ \frac{1}{3} & - \frac{2}{3} \end{matrix}]$ ,

See the step by step solution

Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step by Step Explanation: Step 1: Compute the masses of alloy.

(a)

Consider the different types of coins in terms of equations.

$\begin{array}{rcl} 2 x + 5 y & = & 144 \\ x + y & = & 51 \end{array}$

Represent the equations in terms of matrix and compute the reduced row-echelon form of the matrix.

$\begin{array}{rcl} [\begin{matrix} 2 & 5 & 144 \\ 1 & 1 & 51 \end{matrix}] & = & [\begin{matrix} 2 & 5 & 144 \\ 0 & - \frac{3}{2} & - 21 \end{matrix}] \\ = & [\begin{matrix} 1 & 0 & 37 \\ 0 & 1 & 14 \end{matrix}] \end{array}$

The number of coins is:

$SFr2 = 37 coins, SFr5 = 14 coins$

Step 2: Compute the matrix.

(b)

The matrix that represents the different kinds of coins is,

$[\begin{matrix} number of 2 Franc coins \\ number of 5 Franc coins \end{matrix}] = [\begin{matrix} 2 x \\ 5 y \end{matrix}]$

Consider the mass and volume of alloy in terms of equations.

$\begin{array}{rcl} 2 x + 5 y & = & 144 \\ x + y & = & 51 \end{array}$

Represent the equations in terms of matrix

$[\begin{matrix} 2 & 5 \\ 1 & 1 \end{matrix}] [\begin{matrix} x \\ y \end{matrix}] = [\begin{matrix} 144 \\ 51 \end{matrix}]$

The required matrix is,

$A = [\begin{matrix} 2 & 5 \\ 1 & 1 \end{matrix}]$

The matrix that transforms the different types of coins into the total number and volume of the coins is,

$[\begin{matrix} total value of coins \\ total number of coins \end{matrix}] = [\begin{matrix} 2 & 5 \\ 1 & 1 \end{matrix}]$

Step 3: Check for invertibility and compute the inverse of the matrix.

(c)

Check for the invertible matrix

$|A| = 2 \times 1 - 1 \times 5 \neq 0$

As the determinant of the matrix is nonzero, thus, it is invertible.

The inverse of the matrix is,

role="math" localid="1659777378747" $\begin{array}{rcl} A & = & [\begin{matrix} 2 & 5 \\ 1 & 1 \end{matrix}] \\ \Rightarrow & A^{- 1} = \frac{1}{2 \times 1 - 1 \times 5} [\begin{matrix} 1 & - 5 \\ - 1 & 2 \end{matrix}] \\ \Rightarrow & A^{- 1} = - \frac{1}{3} [\begin{matrix} 1 & - 5 \\ - 1 & 2 \end{matrix}] \\ ∴ A^{- 1} & = & [\begin{matrix} - \frac{1}{3} & \frac{5}{3} \\ \frac{1}{3} & - \frac{2}{3} \end{matrix}] \end{array}$

Hence, the matrix $A$ is invertible and the inverse is $\begin{array}{rcl} A^{- 1} & = & [\begin{matrix} - \frac{1}{3} & \frac{5}{3} \\ \frac{1}{3} & - \frac{2}{3} \end{matrix}] \end{array}$ ,

Find Study Materials for

Create Study Materials

Select your language

Linear Algebra With Applications

Answers without the blur.

Short Answer

Step by Step Solution

Step by Step Explanation: Step 1: Compute the masses of alloy.

Step 2: Compute the matrix.

Step 3: Check for invertibility and compute the inverse of the matrix.

Most popular questions for Math Textbooks

Want to see more solutions like these?

Recommended explanations on Math Textbooks

Select your language

Linear Algebra With Applications

Answers without the blur.

Short Answer

Step by Step Solution

Step by Step Explanation: Step 1: Compute the masses of alloy.

Step 2: Compute the matrix.

Step 3: Check for invertibility and compute the inverse of the matrix.

Most popular questions for Math Textbooks

Want to see more solutions like these?

Recommended explanations on Math Textbooks

Decision Maths

Calculus

Probability and Statistics

Statistics

Mechanics Maths

Geometry

Pure Maths