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Linear Algebra With Applications
Found in: Page 326
Linear Algebra With Applications

Linear Algebra With Applications

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

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Short Answer

Three holy men (let’s call them Anselm, Benjamin, and Caspar) put little stock in material things; their only earthly possession is a small purse with a bit of gold dust. Each day they get together for the following bizarre bonding ritual: Each of them takes his purse and gives his gold away to the two others, in equal parts. For example, if Anselm has 4 ounces one day, he will give 2 ounces each to Benjamin and Caspar.

(a) If Anselm starts out with 6 ounces, Benjamin with 1 ounce, and Caspar with 2 ounces, find formulas for the amounts a(t), b(t), and c(t) each will have after t distributions.

Hint: The vector [111],[1-10] and [10-1], and will be useful.

(b) Who will have the most gold after one year, that is, after 365 distributions?

The solutions are,

at=3.1t+2.-12t+-12t=3+3-12 bt=3-2.-12a ct=3--12

(b) Benjamin

See the step by step solution

Step by Step Solution

Step 1: Solving for (a)

This is dynamical system, with,


This matrix’s eigenvectors are,

v1=111,v2=1-10,v310-1 λ1=1,λ2,3=-12.

With the corresponding eigenvalues being .

We also have



role="math" localid="1659583142610" at=3.1t+2.-12t+-12t=3+3-12t bt=3-2.-12ta ct=3--12t

Step 2: Solving for (b)

We have,

a365=3-312365,b365=3+212365 ,c365=3+12365

So Benjamin will have the most ounces of gold.

Hence final solution is,


(b) Benjamin

at=3.1t+2.-12t+-12t=3+3-12t bt=3-2.-12ta ct=3--12t

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