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In Exercises 62 through 64, consider a function D from to that is linear in both columns and alternating on the columns. See Examples 4 and 6 and the subsequent discussions. Assume that$\mathit{D}\left(I_2\right)\mathbf{=}\mathbf{1}$.

63. Show that$\mathit{D}\left[\begin{array}{cc}a& b\\ 0& d\end{array}\right]\mathbf{=}\mathit{a}\mathit{d}$ . Hint: Write$\left[\begin{array}{c}b\\ d\end{array}\right]\mathbf{=}\left[\begin{array}{c}b\\ 0\end{array}\right]\mathbf{+}\left[\begin{array}{c}0\\ d\end{array}\right]$ and use linearity in the second column: role="math" localid="1660112550370" $\mathit{D}\mathbf{\left[}\begin{array}{cc}\mathbf{a}& \mathbf{b}\\ \mathbf{0}& \mathbf{d}\end{array}\mathbf{\right]}\mathbf{=}\mathit{D}\mathbf{\left[}\begin{array}{cc}\mathbf{a}& \mathbf{b}\\ \mathbf{0}& \mathbf{d}\end{array}\mathbf{\right]}\mathbf{+}\mathit{D}\mathbf{\left[}\begin{array}{cc}\mathbf{a}& \mathbf{0}\\ \mathbf{0}& \mathbf{d}\end{array}\mathbf{\right]}\mathbf{=}\mathit{a}\mathit{b}\mathit{D}\left[\begin{array}{cc}1& 1\\ 0& 0\end{array}\right]\mathbf{+}\mathbf{}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}$ . Use Exercise 62.

Show that more than$\mathit{n}\mathbf{!}\mathbf{=}\mathbf{1}\mathbf{.}\mathbf{2}\mathbf{.}\mathbf{3}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathit{n}$ multiplications are required to compute the determinant of a$\mathbf{n}\mathbf{脳}\mathbf{n}$ matrix by Laplace expansion (for$\mathit{n}\mathbf{>}\mathbf{2}$).

In Exercises 62 through 64, consider a function from to that is linear in both columns and alternating on the columns. See Examples 4 and 6 and the subsequent discussions. Assume that .

64. Using Exercises 62 and 63 as a guide, show that $\mathit{D}\mathbf{\left(}\mathit{A}\mathbf{\right)}\mathbf{=}\mathit{a}\mathit{d}\mathbf{-}\mathit{b}\mathit{c}\mathbf{=}\mathit{d}\mathit{e}\mathit{t}\mathit{A}$for all $\mathbf{2}\mathbf{脳}\mathbf{2}$matrices A .

Show that fewer thane.n! Algebraic operations (additions and multiplications) are required to compute the determinant of a$\mathbf{n}\mathbf{脳}\mathbf{n}$ matrix by Laplace expansion. Hint: Let ${\mathit{L}}_{\mathbf{n}}$is the number of operations required to compute the determinant of a "general"$\mathit{n}\mathbf{脳}\mathit{n}$ matrix by Laplace expansion. Find a formula expressing${\mathit{L}}_{\mathbf{n}}$ in terms of ${\mathit{L}}_{\mathbf{n}\mathbf{-}\mathbf{1}}$. Use this formula to show, by induction (see Appendix B.1), that$\frac{{\mathbf{L}}_{\mathbf{n}}}{\mathbf{n}\mathbf{!}}\mathbf{=}\mathbf{1}\mathbf{+}\mathbf{1}\mathbf{+}\frac{\mathbf{1}}{\mathbf{2}\mathbf{!}}\mathbf{+}\frac{\mathbf{1}}{\mathbf{3}\mathbf{!}}\mathbf{+}\mathit{L}\mathbf{+}\frac{\mathbf{1}}{\left(n-1\right)\mathbf{!}}\mathbf{-}\frac{\mathbf{1}}{\mathbf{n}\mathbf{!}}$ Use the Taylor series of ${\mathit{e}}^{\mathbf{x}}\mathbf{,}{\mathit{e}}^{\mathbf{x}}\mathbf{=}\underset{\mathbf{n}\mathbf{-}\mathbf{0}}{\overset{\mathbf{鈭�}}{\mathbf{鈭�}}}\frac{{\mathbf{x}}^{\mathbf{n}}}{\mathbf{n}\mathbf{!}}$, to show that the right-hand side of this equation is less than e.

Consider a functionDfrom to that is linear in all three columns and alternating on the columns. Assume that $\mathit{D}\left(I_3\right)\mathbf{=}\mathbf{1}$ . Using Exercises 62 through 64 as a guide, show that$\mathit{D}\left(A\right)$ for all$\mathbf{3}\mathbf{脳}\mathbf{3}$ matrices.

Let${\mathit{M}}_{\mathbf{n}}$ be the$\mathit{n}\mathbf{脳}\mathit{n}$ matrix with I's on the main diagonal and directly above the main diagonal,$\mathbf{鈥�}\mathbf{1}\mathbf{\text{'}}\mathbf{s}$directly below the main diagonal, and 0's elsewhere. For example,${\mathit{M}}_{\mathbf{4}}\mathbf{=}\left[\begin{array}{cccc}1& 1& 0& 0\\ 鈥�1& 1& 1& 0\\ 0& 鈥�1& 1& 1\\ 0& 0& 鈥�1& 1\end{array}\right]$

Let ${\mathit{d}}_{\mathbf{n}}\mathbf{=}\mathit{d}\mathit{e}\mathit{t}\left(M_n\right)$ .
a. For $\mathit{n}\mathbf{鈮�}\mathbf{3}$, find a formula expressing${\mathit{d}}_{\mathbf{n}}$ in terms of${\mathit{d}}_{\mathbf{n}\mathbf{-}\mathbf{1}}$ and ${\mathit{d}}_{\mathbf{n}\mathbf{-}\mathbf{2}}$.
b. Find ${\mathit{d}}_{\mathbf{1}}\mathbf{,}{\mathit{d}}_{\mathbf{2}}\mathbf{,}{\mathit{d}}_{\mathbf{3}}\mathbf{,}{\mathit{d}}_{\mathbf{4}}\mathbf{,}\mathbf{}\mathbf{and}\mathbf{}{\mathit{d}}_{\mathbf{10}}$.
c. For which positive integers n is the matrix ${\mathit{M}}_{\mathbf{n}}$invertible?

Let${\mathit{M}}_{\mathbf{n}}$ be therole="math" localid="1660405225698" $\mathit{n}\mathbf{脳}\mathit{n}$ matrix with l's on the main diagonal and directly above the main diagonal,role="math" localid="1660405220979" $\mathbf{-}\mathbf{1}$'s directly below the main diagonal, androle="math" localid="1660405229621" $\mathbf{0}$'s elsewhere. For example ${\mathit{M}}_{\mathbf{4}}\mathbf{=}\left[\begin{array}{cccc}1& 1& 0& 0\\ -1& 1& 1& 0\\ 0& -1& 1& 1\\ 0& 0& -1& 1\end{array}\right]$,

Let ${\mathit{d}}_{\mathbf{n}}\mathbf{=}\mathit{d}\mathit{e}\mathit{t}\left(M_n\right)$

a. For $\mathit{n}\mathbf{猢�}\mathbf{3}$, find a formula expressing ${\mathit{d}}_{\mathbf{n}}$ in terms of${\mathit{d}}_{\mathbf{n}\mathbf{-}\mathbf{1}}$ and${\mathit{d}}_{\mathbf{n}\mathbf{-}\mathbf{2}}$.
b. Find ${\mathit{d}}_{\mathbf{1}}\mathbf{,}{\mathit{d}}_{\mathbf{2}}\mathbf{,}{\mathit{d}}_{\mathbf{3}}\mathbf{,}{\mathit{d}}_{\mathbf{4}}$, and ${\mathit{d}}_{\mathbf{10}}$.
c. For which positive integers$\mathit{n}$ is the matrix${\mathit{M}}_{\mathbf{n}}$ invertible?

Let${\mathit{M}}_{\mathbf{n}}$be the matrix with alllocalid="1660409909298" $\mathbf{1}$'s along the main diagonal, directly above the main diagonal, and directly below the diagonal, and 0's everywhere else. For example, ${\mathit{M}}_{\mathbf{4}}\mathbf{=}\left[\begin{array}{cccc}1& 1& 0& 0\\ 1& 1& 1& 0\\ 0& 1& 1& 1\\ 0& 0& 1& 1\end{array}\right]$

Let localid="1660410059660" ${\mathit{d}}_{\mathbf{n}}\mathbf{=}\mathit{d}\mathit{e}\mathit{t}\mathbf{\left(}{\mathbf{M}}_{\mathbf{n}}\mathbf{\right)}$

a. Find a formula expressinglocalid="1660410070071" ${\mathit{d}}_{\mathbf{n}}$ in terms oflocalid="1660410064552" ${\mathit{d}}_{\mathbf{n}\mathbf{-}\mathbf{1}}$ and localid="1660410074126" ${\mathit{d}}_{\mathbf{n}\mathbf{-}\mathbf{2}}$, for positive integers localid="1660410080136" $\mathit{n}\mathbf{猢�}\mathbf{3}$.
b. Findlocalid="1660410087805" ${\mathit{d}}_{\mathbf{1}}\mathbf{,}{\mathit{d}}_{\mathbf{2}}\mathbf{,}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{,}{\mathit{d}}_{\mathbf{8}}$.
c. What is the relationship betweenlocalid="1660410098480" ${\mathit{d}}_{\mathbf{n}}$ andlocalid="1660410092993" ${\mathit{d}}_{\mathbf{n}\mathbf{+}\mathbf{3}}$? What aboutlocalid="1660410102711" ${\mathit{d}}_{\mathbf{n}}$ and localid="1660410109620" ${\mathit{d}}_{\mathbf{n}\mathbf{+}\mathbf{6}}$?
d. Findlocalid="1660410114201" ${\mathit{d}}_{\mathbf{100}}$

a. LetVbe the linear space of all functionsF from${\mathbf{鈩�}}^{\mathbf{2}\mathbf{脳}\mathbf{2}}$ to$\mathbf{鈩�}$ that are linear in both columns. Find a basis of V, and thus determine the dimension ofV.
b. LetWbe the linear space of all functionsDfrom${\mathbf{鈩�}}^{\mathbf{2}\mathbf{脳}\mathbf{2}}$ to$\mathbf{鈩�}$ that are linear in both columns and alternating on the columns. Find a basis ofW, and thus determine the dimension ofW.