91Ó°ÊÓ

TRUE OR FALSE?

If A is an invertible$\mathbf{2}\mathbf{×}\mathbf{2}$ matrix and B is any $\mathbf{2}\mathbf{×}\mathbf{2}$ matrix, then the formula rref (AB) = rref (B)must hold.

Question: In Exercises 55 through 65, show that the given matrix A is invertible, and find the inverse. Interpret the linear transformation$\mathit{T}\left(\stackrel{→}{x}\right)\mathbf{=}\mathit{A}\stackrel{\mathbf{→}}{\mathbf{x}}$and the inverse transformation${\mathit{T}}^{\mathbf{-}\mathbf{1}}\left(\stackrel{→}{y}\right)\mathbf{=}{\mathbf{A}}^{\mathbf{-}\mathbf{1}}\stackrel{\mathbf{→}}{\mathbf{y}}$ geometrically. Interpret detA geometrically. In your figure, show the angleθ and the vectors$\stackrel{\mathbf{→}}{\mathbf{v}}$ and$\stackrel{→}{\mathit{w}}$ introduced in Theorem 2.4.10.

60.$\left[\begin{array}{cc}-0.8& 0.6\\ 0.6& 0.8\end{array}\right]$.

In Exercises 55 through 64, find all matricesX that satisfy the given matrix equation.

$\left[\begin{array}{ccc}1& 2& 3\\ 0& 1& 2\end{array}\right]X=l_2$

Consider a larger currency exchange matrix (see Exercise 60), involving four of the world’s leading currencies: Euro (C), U.S. dollar (\(), Chinese yuan (¥), and British pound (£).

The entry ${\mathit{a}}_{\mathbf{i}\mathbf{j}}$ gives the value of one unit of the $\mathit{j}\mathit{t}\mathit{h}$ currency, expressed in terms of the $\mathit{i}\mathit{t}\mathit{h}$ currency. For example, ${\mathit{a}}_{\mathbf{34}}\mathbf{=}\mathbf{10}$ means that £1 = ¥10 (as of August 2012). Find the exact values of the 13missing entries ofA(expressed as fractions).

role="math" localid="1664250730664" $\begin{array}{l}\begin{array}{cccc}\mathbf{\text{C}}& \mathbf{\backslash )}& \mathbf{Â\yen }& \mathbf{Â\pounds }\end{array}\\ \mathbf{A}\mathbf{=}\left[\begin{array}{cccc}*& 0.8& *& *\\ *& *& *& *\\ *& *& *& 10\\ 0.8& *& *& *\end{array}\right]\begin{array}{c}\mathbf{C}\\ \mathbf{\$}\\ \mathbf{Â\yen }\\ \mathbf{Â\pounds }\end{array}\end{array}$

TRUE OR FALSE?

There is a transition matrix A such that$\underset{\mathbf{m}\mathbf{→}\mathbf{∞}}{\mathbf{l}\mathbf{i}\mathbf{m}}{\mathit{A}}^{\mathbf{m}}$ fails to exist.

Question: In Exercises 55 through 65, show that the given matrix Ais invertible, and find the inverse. Interpret the linear transformation $\mathit{T}\left(\stackrel{→}{x}\right)\mathbf{=}\mathit{A}\stackrel{\mathbf{→}}{\mathbf{x}}$and the inverse transformation ${\mathit{T}}^{\mathbf{-}\mathbf{1}}\left(\stackrel{→}{y}\right)\mathbf{=}{\mathbf{A}}^{\mathbf{-}\mathbf{1}}\stackrel{\mathbf{→}}{\mathbf{y}}$ geometrically. Interpret det A geometrically. In your figure, show the angle θ and the vectors $\stackrel{\mathbf{→}}{\mathbf{v}}$and $\stackrel{\mathbf{→}}{\mathbf{w}}$introduced in Theorem 2.4.10.

61..$\left[\begin{array}{cc}1& 1\\ -1& 1\end{array}\right]$

TRUE OR FALSE?

For every transition matrix A there exists a nonzero vector $\stackrel{\mathbf{→}}{\mathbf{x}}$ such that$\mathit{A}\stackrel{\mathbf{→}}{\mathbf{x}}\mathbf{=}\stackrel{\mathbf{→}}{\mathbf{x}}$.

Question: In Exercise 55 through 65, show that the given matrix Ais invertible, and find the inverse. Interpret the linear transformation ${\mathit{T}}^{\mathbf{−}\mathbf{1}}\left(\stackrel{→}{x}\right)\mathbf{=}\mathit{A}\stackrel{\mathbf{→}}{\mathbf{x}}$and the inverse transformation ${\mathit{T}}^{\mathbf{−}\mathbf{1}}\left(\stackrel{→}{y}\right)\mathbf{=}{\mathbf{A}}^{\mathbf{−}\mathbf{1}}\stackrel{\mathbf{→}}{\mathbf{y}}$ geometrically. Interpret det Ageometrically. In your figure, show the angle θand the vectors $\stackrel{\mathbf{→}}{\mathbf{v}}\mathbf{}\mathit{a}\mathit{n}\mathit{d}\mathbf{}\stackrel{\mathbf{→}}{\mathbf{w}}$introduced in Theorem 2.4.10.

62.$\left[\begin{array}{cc}1& −1\\ 0& 1\end{array}\right]$

In Exercises 55 through 64, find all matricesX that satisfy the given matrix equation.

$\left[\begin{array}{cc}1& 0\\ 2& 1\\ 3& 3\end{array}\right]X=l_3$

Solving a linear system $\mathit{A}\stackrel{\mathbf{→}}{\mathbf{x}}\mathbf{=}\stackrel{\mathbf{→}}{\mathbf{0}}$by Gaussian elimination amounts to writing the vector of leading variables as a linear transformation of the vector of free variables. Consider the linear system.

$\begin{array}{c}{\mathbf{x}}_{\mathbf{1}}\mathbf{−}{\mathbf{x}}_{\mathbf{2}}\mathbf{+}\mathbf{4}{\mathbf{x}}_{\mathbf{5}}\mathbf{=}\mathbf{0}\\ {\mathbf{x}}_{\mathbf{3}}\mathbf{−}{\mathbf{x}}_{\mathbf{5}}\mathbf{=}\mathbf{0}\\ {\mathbf{x}}_{\mathbf{4}}\mathbf{−}\mathbf{2}{\mathbf{x}}_{\mathbf{5}}\mathbf{=}\mathbf{0}\end{array}$

Find the matrix Bsuch that$\left[\begin{array}{l}{x}_1\\ x_3\\ x_4\end{array}\right]\mathbf{=}\mathit{B}\left[\begin{array}{l}{x}_2\\ x_5\end{array}\right]$.