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Carl’s Sasquatch Attack! Game Card Collection is a mixture of common and rare cards. Each common card is worth \(\$0.25\) while
each rare card is worth \(\$0.75\). If his entire 117 card collection is worth \(\$48.75\), how many of each kind of card does he
own?
Carl owns 78 common cards and 39 rare cards.
How much of a 5 gallon \(40\%\) salt solution should be replaced with pure water to obtain 5 gallons of a \(15 \%\) solution?
\(\answer {3.125}\) gallons.
How much of a 10 liter \(30\%\) acid solution must be replaced with pure acid to obtain 10 liters of a \(50\%\) solution?
\(\frac {20}{7} \approx 2.85\) liters.
Daniel’s Exotic Animal Rescue houses snakes, tarantulas and scorpions. When asked how many animals of each kind he
boards, Daniel answered: ‘We board 49 total animals, and I am responsible for each of their 272 legs and 28 tails.’ How many
of each animal does the Rescue board? (Recall: tarantulas have 8 legs and no tails, scorpions have 8 legs and one tail, and
snakes have no legs and one tail.)
The rescue houses 15 snakes, 21 tarantulas and 13 scorpions.
This exercise is a continuation of Exercise SasquatchDiet in Section MatMethods. Just because a system is consistent independent doesn’t mean it will
admit a solution that makes sense in an applied setting. Using the nutrient values given for Ippizuti Fish, Misty Mushrooms,
and Sun Berries, use Cramer’s Rule to determine the number of servings of Ippizuti Fish needed to meet the needs of
a daily diet which requires 2500 calories, 1000 grams of protein, and 400 milligrams of Vitamin X. Now use
Cramer’s Rule to find the number of servings of Misty Mushrooms required. Does a solution to this diet problem
exist?
Using Cramer’s Rule, we find we need 53 servings of Ippizuti Fish to satisfy the dietary requirements. The number of
servings of Misty Mushrooms required, however, is \(-1120\). Since it’s impossible to have a negative number of servings, there is no
solution to the applied problem, despite there being a solution to the mathematical problem. A cautionary tale about using
Cramer’s Rule: just because you are guaranteed a mathematical answer for each variable doesn’t mean the solution will make
sense in the ‘real’ world.
This exercise is a follow-up to Exercise threepointsmatrixfunctionfitex in Section AugMatrices. Suppose you wish to determine coefficients \(a\), \(b\), and \(c\) so the the graph of \(f(x) = ax^{m} + bx^{n} + cx^{p}\)
contains the points \((-2,1)\), \((1,4)\), \((3,-2)\). With help from your classmates, discuss if there a unique solution for every selection of \(m\), \(n\), and \(p\)? If not,
under what conditions is there a unique solution?
Let \(A\) be an arbitrary invertible \(3 \times 3\) matrix.
Show that \(\det (I_3) = 1\). (See footnote (If you think about it for just a moment, you’ll see that \(\det (I_{n}) = 1\) for any natural number
\(n\). The formal proof of this fact requires the Principle of Mathematical Induction (Section Induction) so we’ll stick with \(n = 3\) for
the time being.) below.)
Using the facts that \(AA^{-1} = I_{3}\) and \(\det (AA^{-1}) = \det (A)\det (A^{-1})\), show that
\[\det (A^{-1}) = \frac {1}{\det (A)}\]
The purpose of Exercises eigenfirst - eigenlast is to introduce you to the eigenvalues and eigenvectors of a matrix. (This material is
usually given its own chapter in a Linear Algebra book so clearly we’re not able to tell you everything you need to know about
eigenvalues and eigenvectors. They are a nice application of determinants, though, so we’re going to give you enough
background so that you can start playing around with them.) We begin with an example using a \(2 \times 2\) matrix and then
guide you through some exercises using a \(3 \times 3\) matrix. Consider the matrix
from Exercise matrixC. We know that \(\det (C) = 0\) which
means that \(CX = 0_{2 \times 2}\) does not have a unique solution. So there is a nonzero matrix \(Y\) with \(CY = 0_{2 \times 2}\). In fact, every matrix of the
form
for all \(t\). The big question is “How did we know to use
\(41\)?”
We need a number \(\lambda \) such that \(CX = \lambda X\) has nonzero solutions. We have demonstrated that \(\lambda = 0\) and \(\lambda = 41\) both worked. Are there others? If we
look at the matrix equation more closely, what we really wanted was a nonzero solution to \((C - \lambda I_2)X = 0_{2 \times 2}\) which we know exists if and only if
the determinant of \(C - \lambda I_2\) is zero. (Think about this.) So we computed
This is called the characteristic polynomial of the
matrix \(C\) and it has two zeros: \(\lambda = 0\) and \(\lambda = 41\). That’s how we knew to use \(41\) in our work above. The fact that \(\lambda = 0\) showed up
as one of the zeros of the characteristic polynomial just means that \(C\) itself had determinant zero which we
already knew. Those two numbers are called the eigenvalues of \(C\). The corresponding matrix solutions to \(CX = \lambda X\) are
called the eigenvectors of \(C\) and the ‘vector’ portion of the name will make more sense after you’ve studied
vectors.
Now it’s your turn. In the following exercises, you’ll be using the matrix \(G\) from Exercise matrixG.
Show that the characteristic polynomial of \(G\) is \(p(\lambda ) = -\lambda (\lambda - 1)(\lambda - 22)\). That is, compute \(\text {det}\left (G - \lambda I_3\right )\).
Let \(G_0 = G\). Find the parametric description of the solution to the system of linear equations given by \(GX = 0_{3 \times 3}\).
Let \(G_1 = G - I_3\). Find the parametric description of the solution to the system of linear equations given by \(G_1X = 0_{3 \times 3}\). Show that any solution to \(G_1X = 0_{3 \times 3}\)
also has the property that \(GX = 1X\).
Let \(G_{22} = G - 22 I_3\). Find the parametric description of the solution to the system of linear equations given by \(G_{22}X = 0_{3 \times 3}\). Show that any solution to \(G_{22}X = 0_{3 \times 3}\)
also has the property that \(GX = 22X\).
