91Ó°ÊÓ

Find the determinants of the linear transformations in Exercises $\mathbf{17}$through .

28.$\mathit{T}\mathbf{\left(}\stackrel{\mathbf{→}}{\mathbf{v}}\mathbf{\right)}\mathbf{=}\left[\begin{array}{l}1\\ 2\\ 3\end{array}\right]\mathbf{×}\stackrel{\mathbf{→}}{\mathbf{v}}$ from the plane$\mathit{V}$ given by ${\mathbf{x}}_{\mathbf{1}}\mathbf{+}\mathbf{2}{\mathbf{x}}_{\mathbf{2}}\mathbf{+}\mathbf{3}{\mathbf{x}}_{\mathbf{3}}\mathbf{=}\mathbf{0}$to $\mathbf{V}$.

a. For an invertible$\mathbf{n}\mathbf{×}\mathbf{n}$ matrix$\mathit{A}$ and an arbitrary$\mathbf{n}\mathbf{×}\mathbf{n}$ matrix $\mathbf{B}$, show that $\mathit{r}\mathit{r}\mathit{e}\mathit{f}\mathbf{[}\mathit{A}\mathbf{âˆ\pounds }\mathit{A}\mathit{B}\mathbf{]}\mathbf{=}\left[\begin{array}{cc}{I}_n& âˆ\pounds B\end{array}\right]\mathit{r}\mathit{r}\mathit{e}\mathit{f}\mathbf{[}\mathit{A}\mathbf{âˆ\pounds }\mathit{A}\mathit{B}\mathbf{]}\mathbf{=}\left[\begin{array}{cc}{I}_n& âˆ\pounds B\end{array}\right]$.Hint: The left part of$\mathbf{rref}\mathbf{[}\mathbf{A}\mathbf{âˆ\pounds }\mathbf{AB}\mathbf{]}$ is $\mathbf{rref}\mathbf{\left(}\mathbf{A}\mathbf{\right)}\mathbf{=}{\mathbf{I}}_{\mathbf{n}}$. Write rref $\mathbf{[}\mathit{A}\mathbf{âˆ\pounds }\mathit{A}\mathit{B}\mathbf{]}\mathbf{=}\left[I_nâˆ\pounds M\right]$; we have to show that$\mathbf{M}\mathbf{=}\mathbf{B}$ . To demonstrate this, note that the columns of matrix $\left[\begin{array}{c}B\\ -I_n\end{array}\right]$are in the kernel of$\mathbf{[}\mathbf{A}\mathbf{âˆ\pounds }\mathbf{AB}\mathbf{]}$ and therefore in the kernel of $\left[I_nâˆ\pounds M\right]$.
b. What does the formula $\mathbf{rref}\mathbf{[}\mathbf{A}\mathbf{âˆ\pounds }\mathbf{AB}\mathbf{]}\mathbf{=}\left[\begin{array}{cc}{I}_n& âˆ\pounds B\end{array}\right]$tell you if $\mathbf{B}\mathbf{=}{\mathbf{A}}^{\mathbf{-}\mathbf{1}}$?

Consider two distinct points$\left[\begin{array}{l}{a}_1\\ a_2\end{array}\right]$ and$\left[\begin{array}{l}{b}_1\\ b_2\end{array}\right]$ in the plane. Explain why the solutions$\left[\begin{array}{l}{x}_1\\ x_2\end{array}\right]$ of the equation$\mathit{d}\mathit{e}\mathit{t}\left[\begin{array}{ccc}1& 1& 1\\ x_1& a_1& b_1\\ x_2& a_2& b_2\end{array}\right]\mathbf{=}\mathbf{0}$ form a line and why this line goes through the two points$\left[\begin{array}{l}{a}_1\\ a_2\end{array}\right]$ and $\left[\begin{array}{l}{b}_1\\ b_2\end{array}\right]$.

Consider a $\mathit{n}\mathbf{×}\mathit{n}$ matrix$\mathit{A}\mathbf{=}\left[{\stackrel{→}{v}}_1{\stackrel{→}{v}}_1....{\stackrel{→}{v}}_n\right]$ . What is the relationship between the product $||{\stackrel{→}{v}}_1||||{\stackrel{→}{v}}_2||\mathbf{.}\mathbf{.}\mathbf{.}||{\stackrel{→}{v}}_n||$ and $\mathbf{\left|}\mathbf{detA}\mathbf{\right|}$When is$\mathbf{\left|}\mathbf{detA}\mathbf{\right|}\mathbf{=}\mathbf{||}{\stackrel{\mathbf{→}}{\mathbf{v}}}_{\mathbf{1}}\mathbf{||}\mathbf{||}{\stackrel{\mathbf{→}}{\mathbf{v}}}_{\mathbf{2}}\mathbf{||}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{||}{\stackrel{\mathbf{→}}{\mathbf{v}}}_{\mathbf{n}}\mathbf{||}$ .

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

10. $\left[\begin{array}{ccccc}1& 1& 1& 1& 1\\ 1& 2& 3& 4& 5\\ 1& 3& 6& 10& 15\\ 1& 4& 10& 20& 35\\ 1& 5& 15& 35& 70\end{array}\right]$

For the matrices A in Exercisethroughfind closed formulas for ${\mathit{A}}^{\mathbf{t}}$ where t is an arbitrary positive integer. follow the strategy. outlined in Theoremand illustrated in Example.in Exercisethroughfeel free to use technology.

role="math" localid="1664272585664" $\mathit{A}\mathbf{=}\left[\begin{array}{ccc}1& 1& 1\\ 0& 0& 1\\ 0& 0& 2\end{array}\right]$

An$\mathbf{8}\mathbf{×}\mathbf{8}$matrix fails to be invertible if (and only if) its determinant is nonzero.

Find all 2x2matrices for which $\left[\begin{array}{c}2\\ 3\end{array}\right]$is an eigenvector with associated eigenvalue -1.

The matrix $\left[\begin{array}{ccc}{k}^2& 1& 4\\ k& -1& -2\\ 1& 1& 1\end{array}\right]$ is invertible for all positive constantsk.

Consider a $\mathbf{4}\mathbf{×4}$matrix A with rows${\stackrel{\mathbf{→}}{\mathbf{v}}}_{\mathbf{1}}\mathbf{,}\stackrel{\mathbf{→}}{{\mathbf{v}}_{\mathbf{2}}}\mathbf{,}\stackrel{\mathbf{→}}{{\mathbf{v}}_{\mathbf{3}}}\mathbf{,}\stackrel{\mathbf{→}}{{\mathbf{v}}_{\mathbf{4}}}$. If $\mathit{d}\mathit{e}\mathit{t}\left(A\right)\mathbf{=}\mathbf{8}$, find the determinants in Exercises 11 through 16.

11. $\mathit{d}\mathit{e}\mathit{t}\left[\begin{array}{c}{\stackrel{→}{v}}_1\\ {\stackrel{→}{v}}_2\\ -9{\stackrel{→}{v}}_3\\ {\stackrel{→}{v}}_4\end{array}\right]$