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Found in: Page 57

### Linear Algebra With Applications

Book edition 5th
Author(s) Otto Bretscher
Pages 442 pages
ISBN 9780321796974

# Some parking meters in downtown Geneva, Switzerland, accept ${\mathbf{2}}$ Franc and ${\mathbf{5}}$ Franc coins. a. A parking officer collects ${\mathbf{51}}$ coins worth ${\mathbf{144}}$ Francs. How many coins are there of each kind?b. Find the matrix ${\mathbit{A}}$ that transforms the vector $\left[\begin{array}{c}\text{number of 2 Franc coins}\\ \text{number of 5 Franc coins}\end{array}\right]$into the vector $\left[\begin{array}{c}\text{total value of coins}\\ \text{total number of coins}\end{array}\right]$c. Is the matrix ${\mathbit{A}}$ 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 $SFr2$ and $14$coins of $SFr5$.

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

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

See the step by step solution

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

(a)

Consider the different types of coins in terms of equations.

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

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

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

The number of coins is:

$\text{SFr2}=37 \text{coins},\text{SFr5}=14 \text{coins}$

## Step 2: Compute the matrix.

(b)

The matrix that represents the different kinds of coins is,

$\left[\begin{array}{c}\text{number of 2 Franc coins}\\ \text{number of 5 Franc coins}\end{array}\right]=\left[\begin{array}{c}2x\\ 5y\end{array}\right]$

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

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

Represent the equations in terms of matrix

$\left[\begin{array}{cc}2& 5\\ 1& 1\end{array}\right]\left[\begin{array}{c}x\\ y\end{array}\right]=\left[\begin{array}{c}144\\ 51\end{array}\right]$

The required matrix is,

$A=\left[\begin{array}{cc}2& 5\\ 1& 1\end{array}\right]$

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

$\left[\begin{array}{c}\text{total value of coins}\\ \text{total number of coins}\end{array}\right]=\left[\begin{array}{cc}2& 5\\ 1& 1\end{array}\right]$

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

(c)

Check for the invertible matrix

$\left|A\right|=2×1-1×5\ne 0$

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

The inverse of the matrix is,

role="math" localid="1659777378747" $\begin{array}{rcl}A& =& \left[\begin{array}{cc}2& 5\\ 1& 1\end{array}\right]\\ & ⇒& {A}^{-1}=\frac{1}{2×1-1×5}\left[\begin{array}{cc}1& -5\\ -1& 2\end{array}\right]\\ & ⇒& {A}^{-1}=-\frac{1}{3}\left[\begin{array}{cc}1& -5\\ -1& 2\end{array}\right]\\ \therefore {A}^{-1}& =& \left[\begin{array}{cc}-\frac{1}{3}& \frac{5}{3}\\ \frac{1}{3}& -\frac{2}{3}\end{array}\right]\end{array}$

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

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