91影视

Let ${\mathit{M}}_{\mathbf{n}}$be the$\mathit{n}\mathbf{脳}\mathit{n}$matrix with all 1's along "the other diagonal," and0鈥檚 everywhere else. For example,

${\mathit{M}}_{\mathbf{4}}\mathbf{=}\left[\begin{array}{cccc}0& 0& 0& 1\\ 0& 0& 1& 0\\ 0& 1& 0& 0\\ 1& 0& 0& 0\end{array}\right]$

a. Find$\mathbf{det}\mathbf{\left(}{\mathbf{M}}_{\mathbf{n}}\mathbf{\right)}\mathbf{for}\mathbf{}\mathbf{n}\mathbf{=}\mathbf{2}\mathbf{,}\mathbf{3}\mathbf{,}\mathbf{4}\mathbf{,}\mathbf{5}\mathbf{,}\mathbf{6}\mathbf{,}\mathbf{7}$.
b. Find a formula for$\mathbf{det}\mathbf{\left(}{\mathbf{M}}_{\mathbf{n}}\mathbf{\right)}$, in terms of n.

Consider a linear transformation T from ${\mathbf{R}}^{\mathbf{m}\mathbf{+}\mathbf{n}}$ to ${\mathbf{R}}^{\mathbf{m}}$ . The matrix Aof T can be written in block form as $\mathit{A}\mathit{=}\left[\begin{array}{cc}{A}_1& A_2\end{array}\right]$, where ${\mathit{A}}_{\mathbf{1}}$is $\mathit{m}\mathbf{脳}\mathit{m}$and ${\mathit{A}}_{\mathbf{2}}$is $\mathit{m}\mathbf{脳}\mathit{n}$. Suppose that $\mathit{d}\mathit{e}\mathit{t}\mathbf{\left(}{\mathit{A}}_{\mathbf{1}}\mathbf{\right)}\mathbf{鈮�}\mathbf{0}$. Show that for every vector $\stackrel{\mathbf{鈫�}}{\mathbf{x}}$in ${\mathit{R}}^{\mathbf{n}}$ there exists a unique $\stackrel{\mathbf{鈫�}}{\mathbf{y}}$in ${\mathit{R}}^{\mathbf{m}}$such that $\mathit{T}\left[\begin{array}{c}\stackrel{鈫�}{y}\\ \stackrel{鈫�}{x}\end{array}\right]\mathbf{=}\stackrel{\mathbf{鈫�}}{\mathbf{0}}$.Show that the transformation$\stackrel{\mathbf{鈫�}}{\mathbf{x}}\mathbf{鈫�}\stackrel{\mathbf{鈫�}}{\mathbf{y}}$from ${\mathbf{R}}^{\mathbf{n}}$ to ${\mathbf{R}}^{\mathbf{m}}$ is linear, and find its matrix M (in terms of ${\mathit{A}}_{\mathbf{1}}$and ${\mathit{A}}_{\mathbf{2}}$). (This is the linear version of the implicit function theorem of multivariable calculus.)

A square matrix is called a permutation matrix if each row and each column contains exactly one entry 1, with all other entries being 0 . Examples are ${\mathit{I}}_{\mathbf{n}}\mathbf{,}\mathbf{\left[}\begin{array}{ccc}\mathbf{0}& \mathbf{1}& \mathbf{0}\\ \mathbf{0}& \mathbf{0}& \mathbf{1}\\ \mathbf{1}& \mathbf{0}& \mathbf{0}\end{array}\mathbf{\right]}$, and the matrices considered in Exercises 53 and 56 . What are the possible values of the determinant of a permutation matrix?

Find the matrix Mintroduced in Exercise 57 for the linear transformation$\mathbf{T}\left(\stackrel{鈬赌}{v}\right)\mathbf{=}\left[\begin{array}{cccc}1& 2& 1& 2\\ 3& 7& 4& 3\end{array}\right]\stackrel{\mathbf{鈬赌}}{\mathbf{v}}$You can either follow the approach outlined in Exercise 57 or use Gaussian elimination, expressing the leading variables ${\mathit{y}}_{\mathbf{1}}\mathbf{,}{\mathit{y}}_{\mathbf{2}}$ in terms of the free variableslocalid="1660716914088" ${\mathbf{x}}_{\mathbf{1}}\mathbf{,}{\mathbf{x}}_{\mathbf{2}}\mathbf{}$where$\stackrel{\mathbf{鈬赌}}{\mathbf{v}}\mathbf{=}\left[\begin{array}{c}{y}_1\\ y_2\\ x_1\\ x_2\end{array}\right]$. Note that this procedure amounts to finding the kernel of, in the familiar way; we just interpret the result somewhat differently.

a. Find a noninvertible $\mathbf{2}\mathbf{脳}\mathbf{2}$matrix whose entries are four distinct prime numbers, or explain why no such matrix exists.

b. Find a noninvertible $\mathbf{3}\mathbf{脳}\mathbf{3}$matrix whose entries are nine distinct prime numbers, or explain why no such matrix exists.

If the equation$\mathit{d}\mathit{e}\mathit{t}\mathbf{}\mathbf{A}\mathbf{=}\mathit{d}\mathit{e}\mathit{t}\mathbf{}\mathbf{B}$holds for two n x n matrices A and B , is A necessarily similar to B?

Consider the function $\mathit{F}\left(A\right)\mathbf{=}\mathit{F}\left[\stackrel{鈫�}{v}\stackrel{鈫�}{w}\right]\mathbf{=}\stackrel{\mathbf{鈫�}}{\mathbf{v}}\mathbf{}\mathbf{脳}\mathbf{}\stackrel{\mathbf{鈫�}}{\mathbf{w}}$from ${\mathbf{鈩�}}^{\mathbf{2}\mathbf{脳}\mathbf{2}}$ to$\mathrm{鈩�}$, the dot product of the column vectors of A.
a. Is Flinear in both columns of A? See Example 6.
b. Is F linear in both rows of A?
c. Is Falternating on the columns of A? See Example 4.

The tetrahedron defined by three vectors ${\stackrel{\mathbf{鈫�}}{\mathbf{v}}}_{\mathbf{1}}\mathbf{,}{\stackrel{\mathbf{鈫�}}{\mathbf{v}}}_{\mathbf{2}}\mathbf{,}{\stackrel{\mathbf{鈫�}}{\mathbf{v}}}_{\mathbf{3}}\mathbf{}\mathbf{in}\mathbf{}{\mathit{R}}^{\mathbf{3}}$is the set of all vectors of the form ${\mathit{c}}_{\mathbf{1}}{\stackrel{\mathbf{鈫�}}{\mathbf{v}}}_{\mathbf{1}}\mathbf{+}{\mathit{c}}_{\mathbf{2}}{\stackrel{\mathbf{鈫�}}{\mathbf{v}}}_{\mathbf{2}}\mathbf{+}{\mathit{c}}_{\mathbf{3}}{\stackrel{\mathbf{鈫�}}{\mathbf{v}}}_{\mathbf{3}}$, where ${\mathit{c}}_{\mathbf{i}}\mathbf{鈮�}\mathbf{0}$and ${\mathit{c}}_{\mathbf{1}}\mathbf{+}{\mathit{c}}_{\mathbf{2}}\mathbf{+}{\mathit{c}}_{\mathbf{3}}\mathbf{鈮�}\mathbf{1}$. Explain why the volume of this tetrahedron is one sixth of the volume of the parallelepiped defined by ${\stackrel{\mathbf{鈫�}}{\mathbf{v}}}_{\mathbf{1}}\mathbf{,}{\stackrel{\mathbf{鈫�}}{\mathbf{v}}}_{\mathbf{2}}\mathbf{,}{\stackrel{\mathbf{鈫�}}{\mathbf{v}}}_{\mathbf{3}}$.

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

5.$\mathbf{\left[}\begin{array}{cccc}0& 2& 3& 4\\ 0& 0& 0& 4\\ 1& 2& 3& 4\\ 0& 0& 3& 4\end{array}\mathbf{\right]}$

If$\mathit{A}\mathbf{=}\mathbf{\left[}\stackrel{\mathbf{鈫�}}{\mathbf{u}}\stackrel{\mathbf{鈫�}}{\mathbf{v}}\stackrel{\mathbf{鈫�}}{\mathbf{w}}\mathbf{\right]}$ is any$\mathbf{3}\mathbf{脳}\mathbf{3}$ matrix, then$\mathit{d}\mathit{e}\mathit{t}\mathit{A}\mathbf{=}\stackrel{\mathbf{鈫�}}{\mathbf{u}}\mathbf{.}\mathbf{(}\stackrel{\mathbf{鈫�}}{\mathbf{v}}\mathbf{脳}\stackrel{\mathbf{鈫�}}{\mathbf{w}}\mathbf{)}$ .