91影视

Find the determinant of the$\left(2n\right)\mathbf{脳}\left(2n\right)$matrix

$\mathit{A}\mathbf{=}\left[\begin{array}{cc}0& l_n\\ I_n& 0\end{array}\right]$.

Consider a $\mathbf{2}\mathbf{脳}\mathbf{2}$ matrix$\mathit{A}\mathbf{=}\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]$with column vectors$\stackrel{\mathbf{鈫�}}{\mathbf{v}}\mathbf{=}\left[\begin{array}{c}a\\ c\end{array}\right]$ and $\stackrel{\mathbf{鈫�}}{\mathbf{w}}\mathbf{=}\left[\begin{array}{c}b\\ d\end{array}\right]$ .We define the linear transformation$\mathit{T}\mathbf{\left(}\stackrel{\mathbf{鈫�}}{\mathbf{x}}\mathbf{\right)}\mathbf{=}\left[\begin{array}{c}det\left[\begin{array}{cc}\stackrel{鈫�}{x}& \stackrel{鈫�}{w}\end{array}\right]\\ det\left[\begin{array}{cc}\stackrel{鈫�}{v}& \stackrel{鈫�}{x}\end{array}\right]\end{array}\right]$ from ${\mathit{R}}^{\mathbf{2}}$ to ${\mathit{R}}^{\mathbf{2}}$ .
a. Find the standard matrix $\mathit{B}$ of $\mathit{T}$ . (Write the entries of $\mathit{B}$ in terms of the entries $\mathit{a}\mathbf{,}\mathit{b}\mathbf{,}\mathit{c}\mathbf{,}\mathit{d}$ of $\mathit{A}$.)
b. What is the relationship between the determinants of $\mathit{A}$and $\mathit{B}$?
c. Show that $\mathit{B}\mathit{A}$ is a scalar multiple of ${\mathit{I}}_{\mathbf{2}}$ . What about $\mathit{A}\mathit{B}$?

d. If $\mathit{A}$ is noninvertible (but nonzero), what is the relationship between the image of $\mathit{A}$ and the kernel of $\mathit{B}$ ? What about the kernel of$\mathit{A}$ and the image of $\mathit{B}$ ?
e. If$\mathit{A}$ is invertible, what is the relationship between$\mathit{B}$ and${\mathit{A}}^{\mathbf{-}\mathbf{1}}$ ?

Consider two vectors$\stackrel{\mathbf{鈬赌}}{\mathbf{v}}$ and $\stackrel{\mathbf{鈬赌}}{\mathbf{w}}$in${\mathbf{鈩�}}^{\mathbf{3}}$. Form the matrix $\mathit{A}\mathbf{=}\left[\stackrel{鈬赌}{v}x\stackrel{鈬赌}{w}\stackrel{鈬赌}{v}\stackrel{鈬赌}{w}\right]$ . Express detA in terms of$||\stackrel{鈬赌}{v}x\stackrel{鈬赌}{w}||$. For which choices of $\stackrel{\mathbf{鈬赌}}{\mathbf{v}}$ and $\stackrel{\mathbf{鈬赌}}{\mathbf{w}}$ is Ainvertible?

Consider an invertible$\mathbf{2}\mathbf{脳}\mathbf{2}$matrix $\mathit{A}$ with integer entries.
a. Show that if the entries of${\mathit{A}}^{\mathbf{-}\mathbf{1}}$are integers, thenlocalid="1660721234455" $\mathit{d}\mathit{e}\mathit{t}\mathit{A}\mathbf{=}\mathbf{1}$or $\mathit{d}\mathit{e}\mathit{t}\mathit{A}\mathbf{=}\mathbf{-}\mathbf{1}$.
b. Show the converse: If $\mathit{d}\mathit{e}\mathit{t}\mathit{A}\mathbf{=}\mathbf{1}$ or $\mathit{d}\mathit{e}\mathit{t}\mathit{A}\mathbf{=}\mathbf{-}\mathbf{1}$, then the entries of ${\mathit{A}}^{\mathbf{-}\mathbf{1}}$are integers.

Find the determinant of the (2n) x (2n)matrix$\mathit{A}\mathbf{=}\left[\begin{array}{cc}0& I_n\\ I_n& 0\end{array}\right]$

Let Aand B bea$\mathbf{2}\mathbf{脳}\mathbf{2}$ matrices with integer entries such that $\mathit{A}\mathbf{,}\mathit{A}\mathbf{+}\mathit{B}\mathbf{,}\mathit{A}\mathbf{+}\mathbf{2}\mathit{B}\mathbf{,}\mathit{A}\mathbf{+}\mathbf{3}\mathit{B}$ , and $\mathit{A}\mathbf{+}\mathbf{4}\mathit{B}$are all invertible matrices whose inverses have integer entries. Show that $\mathit{A}\mathbf{+}\mathbf{5}\mathit{B}$ is invertible and that it鈥檚 inverse has integer entries. This question was in the William Lowell Putnam Mathematical Competition in 1994. Hint: Consider the function $\mathit{f}\mathbf{\left(}\mathit{t}\mathbf{\right)}\mathbf{=}\mathbf{\left(}\mathit{d}\mathit{e}\mathit{t}\mathbf{\right(}\mathit{A}\mathbf{+}\mathit{t}\mathit{B}\mathbf{)}{\mathbf{)}}^{\mathbf{2}}\mathbf{-}\mathbf{1}$. Shows that this is a polynomial; what can you say about its degree? Find the values $\mathit{f}\mathbf{\left(}\mathbf{0}\mathbf{\right)}\mathbf{,}\mathit{f}\mathbf{\left(}\mathbf{1}\mathbf{\right)}\mathbf{,}\mathit{f}\mathbf{\left(}\mathbf{2}\mathbf{\right)}\mathbf{,}\mathit{f}\mathbf{\left(}\mathbf{3}\mathbf{\right)}\mathbf{,}\mathit{f}\mathbf{\left(}\mathbf{4}\mathbf{\right)}$, using Exercise 53. Now you can determine $\mathit{f}\left(t\right)$by using a familiar result: If a polynomial $\mathit{f}\left(t\right)$of degree $\mathbf{鈮�}\mathit{m}$has more than zeros, then$\mathit{f}\mathbf{\left(}\mathit{t}\mathbf{\right)}\mathbf{=}\mathbf{0}$ for all t.

Is the determinant of the matrix

$\mathit{A}\mathbf{=}\left[\begin{array}{ccccc}1& 1000& 2& 3& 4\\ 5& 6& 7& 1000& 8\\ 1000& 9& 8& 7& 6\\ 5& 4& 3& 2& 1000\\ 1& 2& 1000& 3& 4\end{array}\right]$

positive or negative? How can you tell? Do not use technology.

For a fixed positive integer $\mathit{n}$ , let$\mathit{D}$ be a function which assigns to any $\mathit{n}\mathbf{脳}\mathit{n}$matrix$\mathit{A}$ a number $\mathit{D}\mathbf{\left(}\mathit{A}\mathbf{\right)}$such that
a. $\mathit{D}$ is linear in the rows (see Theorem 6.2.2),
b. $\mathit{D}\mathbf{\left(}\mathit{B}\mathbf{\right)}\mathbf{=}\mathbf{-}\mathit{D}\mathbf{\left(}\mathit{A}\mathbf{\right)}$ if$\mathit{B}$ is obtained from$\mathit{A}$ by a row swap, and
c. .$\mathit{D}\mathbf{\left(}{\mathit{I}}_{\mathbf{n}}\mathbf{\right)}\mathbf{=}\mathbf{1}$
Show that$\mathit{D}\mathbf{\left(}\mathit{A}\mathbf{\right)}\mathbf{=}\mathit{d}\mathit{e}\mathit{t}\mathbf{\left(}\mathit{A}\mathbf{\right)}$ for all$\mathit{n}\mathbf{脳}\mathit{n}$ matrices$\mathit{A}$ .

Does the following matrix have an LU factorization? See Exercises 2.4.90 and 2.4.93.

$\mathit{A}\mathbf{=}\left[\begin{array}{ccc}7& 4& 2\\ 5& 3& 1\\ 3& 1& 4\end{array}\right]$

Use the characterization of the determinant given in Exercise 55 to show that $\mathit{d}\mathit{e}\mathit{t}\mathbf{\left(}\mathit{A}\mathit{M}\mathbf{\right)}\mathbf{=}\mathbf{\left(}\mathit{d}\mathit{e}\mathit{t}\mathit{A}\mathbf{\right)}\mathbf{\left(}\mathit{d}\mathit{e}\mathit{t}\mathit{M}\mathbf{\right)}$.