91影视

Question: Using the terminology introduced in the proof of Theorem 6.2.10, show that

sgn P = (-1)^i+j sgn(P_ij). See Exercise 67.

Question: Let G be the set of all integers x that can be written as the sum of the squares of two integers, x = a² + b² . For example, 13 = 3² + 2² is in G , while 7 fails to be in G .
a. List all integers x< 10 that are in G .
b. Show that G is closed under multiplication: If x = a² + b² and y = c² + d² are in G , then so is their product . Hint: Consider the matrices , ,their product, and their determinants.

c. Given that 2642 = 31¹ + 41² and 3218 = 37²+ 43² , write 8,501,956 2642.3218 as the sum of the squares of two positive integers. You may use technology.

In Exercises 5 through 40, find the matrix of the given linear transformation with respect to the given basis. If no basis is specified, use standard basis:for,

forandfor,.For the spaceof upper triangularmatrices, use the basis

Unless another basis is given. In each case, determine whetheris an isomorphism. Ifisn鈥檛 an isomorphism, find bases of the kernel and image ofand thus determine the rank of.

21. from to with respect to the basis.

Consider $\mathit{n}\mathbf{脳}\mathit{n}$ matrices$\mathit{A}\mathbf{,}\mathit{B}\mathbf{,}\mathit{C}$, and$\mathit{D}$ such that $\mathit{r}\mathit{a}\mathit{n}\mathit{k}\mathbf{\left(}\mathit{A}\mathbf{\right)}\mathbf{=}\mathit{r}\mathit{a}\mathit{n}\mathit{k}\left[\begin{array}{cc}A& B\\ C& D\end{array}\right]\mathbf{=}\mathit{n}$. Show that

a. $\mathit{D}\mathbf{=}\mathit{C}{\mathit{A}}^{\mathbf{-}\mathbf{1}}\mathit{B}$, and
b. The$\mathbf{2}\mathbf{脳}\mathbf{2}$ matrix$\left[\begin{array}{cc}det\left(A\right)& det\left(B\right)\\ det\left(C\right)& det\left(D\right)\end{array}\right]$ is noninvertible. Hint: Consider the product $\left[\begin{array}{cc}{I}_n& 0\\ -C{A}^{-1}& I_n\end{array}\right]\left[\begin{array}{cc}A& B\\ C& D\end{array}\right]$.

ConsiderNXN matricesA,B,C and D such that $\mathit{r}\mathit{a}\mathit{n}\mathit{k}\left(A\right)\mathbf{=}\mathit{r}\mathit{a}\mathit{n}\mathit{k}\left[\begin{array}{cc}A& B\\ C& D\end{array}\right]\mathbf{=}\mathit{n}$. Show that
a.$\mathit{D}\mathbf{=}\mathit{C}{\mathit{A}}^{\mathbf{-}\mathbf{1}}\mathit{B}$and
b. The 2X12 matrix$\left[\begin{array}{cc}\det \left(A\right)& \det \left(B\right)\\ \det \left(C\right)& \det \left(D\right)\end{array}\right]$ is noninvertible. Hint: Consider the product $\left[\begin{array}{cc}{I}_n& 0\\ -C{A}^{-1}& I_n\end{array}\right]\left[\begin{array}{cc}A& B\\ C& D\end{array}\right]$.

In Exercises 62 through 64, consider a function D from ${\mathbf{鈩�}}^{\mathbf{2}\mathbf{脳}\mathbf{2}}$ to$\mathbf{鈩�}$ that is linear in both columns and alternating on the columns. See Examples 4 and 6 and the subsequent discussions. Assume that$\mathit{D}\mathbf{\left(}{\mathit{l}}_{\mathbf{2}}\mathbf{\right)}\mathbf{=}\mathbf{1}$.

62. Show that$\mathit{D}\mathbf{\left(}\mathit{A}\mathbf{\right)}\mathbf{=}\mathbf{0}$for any$\mathbf{2}\mathbf{脳}\mathbf{2}$matrix whose two columns are equal.

Question: Consider those 4 脳 4 matrices whose entries are all1,-1, or0. What is the maximal value of the determinant of a matrix of this type? Give an example of a matrix whose determinant has this maximal value.