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In an economics text${}^{\mathbf{11}}$we find the following system:

$\left[\begin{array}{ccc}-R_1& R_1& -\left(1-伪\right)\\ 伪& 1-伪& -{\left(1-伪\right)}^2\\ R_2& -R_2& \frac{-{\left(1-伪\right)}^2}{伪}\end{array}\right]\left[\begin{array}{c}d{x}_1\\ d{y}_1\\ dp\end{array}\right]\mathbf{=}\left[\begin{array}{c}0\\ 0\\ -R_{2}d{e}_2\end{array}\right]$.

Solve for $\mathit{d}{\mathit{x}}_{\mathbf{1}}\mathbf{,}\mathit{d}{\mathit{y}}_{\mathbf{1}}$, and dp. In your answer, you may refer to the determinant of the coefficient matrix as D. (You need not compute D.) The quantities${\mathit{R}}_{\mathbf{1}}\mathbf{,}{\mathit{R}}_{\mathbf{2}}$and D are positive, and ais between zero and one. If $\mathit{d}{\mathit{e}}_{\mathbf{2}}$is positive, what can you say about the signs of $\mathit{d}{\mathit{y}}_{\mathbf{1}}$and dp?

Let ${\mathit{P}}_{\mathbf{n}}$ be the $\mathit{n}\mathbf{脳}\mathit{n}$ matrix whose entries are all ones, except for zeros directly below the main diagonal; for example,

role="math" localid="1659508976827" ${\mathit{P}}_{\mathbf{5}}\mathbf{\left[}\begin{array}{ccccc}\mathbf{1}& \mathbf{1}& \mathbf{1}& \mathbf{1}& \mathbf{1}\\ \mathbf{0}& \mathbf{1}& \mathbf{1}& \mathbf{1}& \mathbf{1}\\ \mathbf{1}& \mathbf{0}& \mathbf{1}& \mathbf{1}& \mathbf{1}\\ \mathbf{1}& \mathbf{1}& \mathbf{0}& \mathbf{1}& \mathbf{1}\\ \mathbf{1}& \mathbf{1}& \mathbf{1}& \mathbf{0}& \mathbf{1}\end{array}\mathbf{\right]}$

Find the determinant of ${\mathit{P}}_{\mathbf{n}}$ .

29. If$A=\left[\stackrel{鈫�}{u}\stackrel{鈫�}{v}\stackrel{鈫�}{w}\right]$is a$3脳3$ matrix, then the formula$det\left(A\right)=\stackrel{鈫�}{V}.\left(\stackrel{鈫�}{u}脳\stackrel{鈫�}{w}\right)$ must hold.

If$\mathit{d}\mathit{e}\mathit{t}\left(A^{10}\right)\mathbf{=}\mathbf{(}\mathbf{d}\mathbf{e}\mathbf{t}\mathbf{A}{\mathbf{)}}^{\mathbf{10}}$for all $\mathbf{10}\mathbf{脳}\mathbf{10}$matrices$\mathit{A}$.

Find the area of the triangle defined by $\left[\begin{array}{c}3\\ 7\end{array}\right]$and$\left[\begin{array}{c}8\\ 2\end{array}\right]$.

Use Gaussian elimination to find the determinant of the matrices A in Exercises 1 through 10.

2.$\left[\begin{array}{ccc}1& 2& 3\\ 1& 6& 8\\ -2& -4& 0\end{array}\right]$

There exist invertible$\mathbf{2}\mathbf{脳}\mathbf{2}$matrices A andBsuch that$\mathit{d}\mathit{e}\mathit{t}\left(A+B\right)\mathbf{=}\mathit{d}\mathit{e}\mathit{t}\mathit{A}\mathbf{+}\mathit{d}\mathit{e}\mathit{t}\mathit{B}$ .

Consider two distinct real numbers, a and b. We define the function

$\mathit{f}\left(t\right)\mathbf{=}\mathit{d}\mathit{e}\mathit{t}\left[\begin{array}{ccc}1& 1& 1\\ a& b& t\\ a^2& b^2& t^2\end{array}\right]$

a. Show that$\mathbf{f}\left(t\right)$ is a quadratic function. What is the coefficient of${\mathit{t}}^{\mathbf{2}}$?
b. Explain why$\mathit{f}\left(a\right)\mathbf{=}\mathit{f}\left(b\right)\mathbf{=}\mathbf{0}$. Conclude that$\mathit{f}\left(t\right)\mathbf{=}\mathit{k}\left(t-a\right)\left(t-b\right)$, for some constant k. Find k, using your work in part (a).
c. For which values of tis the matrix invertible?

Vandermonde determinants (introduced by Alexandre-Th茅ophile Vandermonde). Consider distinct real numbers ${\mathit{a}}_{\mathbf{0}}\mathbf{,}{\mathit{a}}_{\mathbf{1}}\mathbf{,}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{,}{\mathit{a}}_{\mathbf{n}\mathbf{.}}$. We define$\left(n+1\right)\mathbf{脳}\left(n+1\right)$ the matrix

$\mathit{A}\mathbf{=}\left[\begin{array}{ccc}1& 1& ....1\\ a_0& a_1& ....a_n\\ a_0^2& a_1^2& ....a_1^2\\ a_0^n& a_1^n& ....a_n^n\end{array}\right]$

Vandermonde showed that

$\mathit{d}\mathit{e}\mathit{t}\mathbf{\left(}\mathbf{A}\mathbf{\right)}\mathbf{=}\underset{\mathbf{i}\mathbf{>}\mathbf{j}}{\mathbf{鈭�}}\mathbf{(}{\mathbf{a}}_{\mathbf{i}}\mathbf{-}{\mathbf{a}}_{\mathbf{j}}\mathbf{)}$

the product of all differences$\left(a_i-a_j\right)$, where exceeds j.
a. Verify this formula in the case of$\mathit{n}\mathbf{=}\mathbf{1}$.
b. Suppose the Vandermonde formula holds for$\mathit{n}\mathbf{=}\mathbf{1}$. You are asked to demonstrate it for n. Consider the function

$\mathbf{f}\left(t\right)\mathbf{=}\mathbf{det}\left[\begin{array}{ccccc}1& 1& ...& 1& 1\\ a_0& a_1& ...& a_{n-1}& t\\ a_0^2& a_1^2& ...& a_{n-1}& t^2\\ 鈰�& 鈰�& ...& 鈰�& 鈰�\\ a_0^n& a_1^n& ...& a_{n-1}^n& t^n\end{array}\right]$

Explain why f(t) is a polynomial of ${\mathit{n}}^{\mathbf{t}\mathbf{h}}$degree. Find the coefficient k of${\mathit{t}}^{\mathbf{n}}$ using Vandermonde's formula for${\mathit{a}}_{\mathbf{0}}\mathbf{,}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{,}{\mathit{a}}_{\mathbf{n}\mathbf{-}\mathbf{1}}$. Explain why

role="math" localid="1659522435181" $\mathit{f}\mathbf{\left(}{\mathbf{a}}_{\mathbf{0}}\mathbf{\right)}\mathbf{=}\mathit{f}\mathbf{\left(}{\mathbf{a}}_{\mathbf{1}}\mathbf{\right)}\mathbf{=}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{=}\mathit{f}\mathbf{\left(}{\mathbf{a}}_{\mathbf{n}\mathbf{-}\mathbf{1}}\mathbf{\right)}\mathbf{=}\mathbf{0}$

Conclude that

$\mathit{f}\mathbf{\left(}\mathit{t}\mathbf{\right)}\mathbf{=}\mathit{k}\mathbf{(}\mathit{t}\mathbf{-}{\mathbf{a}}_{\mathbf{0}}\mathbf{)}\mathbf{(}\mathit{t}\mathbf{-}{\mathbf{a}}_{\mathbf{1}}\mathbf{)}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{(}\mathit{t}\mathbf{-}{\mathbf{a}}_{\mathbf{n}\mathbf{-}\mathbf{1}}\mathbf{)}$

for the scalar k you found above. Substitute$\mathbf{t}\mathbf{=}{\mathbf{a}}_{\mathbf{n}}$ to demonstrate Vandermonde's formula.

There exist real invertible $\mathbf{3}\mathbf{脳}\mathbf{3}$ matrices A andSsuch that${\mathit{S}}^{\mathbf{-}\mathbf{1}}\mathit{A}\mathit{S}\mathbf{=}\mathbf{2}\mathit{A}$ .