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

Consider a linear transformation $\mathit{T}\left(\stackrel{鈫�}{x}\right)\mathbf{=}\mathit{A}\stackrel{\mathbf{鈫�}}{\mathbf{x}}$ from ${\mathbf{鈩�}}^{\mathbf{2}}$to${\mathbf{鈩�}}^{\mathbf{2}}$ . Suppose for two vectors ${\stackrel{\mathbf{鈫�}}{\mathbf{v}}}_{\mathbf{1}}$ and ${\stackrel{\mathbf{鈫�}}{\mathbf{v}}}_{\mathbf{2}}$in ${\mathbf{鈩�}}^{\mathbf{2}}$ we have $\mathit{T}\mathbf{\left(}{\stackrel{\mathbf{鈫�}}{\mathbf{v}}}_{\mathbf{1}}\mathbf{\right)}\mathbf{=}\mathbf{3}{\stackrel{\mathbf{鈫�}}{\mathbf{v}}}_{\mathbf{1}}$androle="math" localid="1660719222222" $\mathit{T}\mathbf{\left(}{\stackrel{\mathbf{鈫�}}{\mathbf{v}}}_{\mathbf{2}}\mathbf{\right)}\mathbf{=}\mathbf{4}{\stackrel{\mathbf{鈫�}}{\mathbf{v}}}_{\mathbf{2}}$ . What can you say about det A ? Justify your answer carefully.

Consider those $\mathbf{4}\mathbf{脳}\mathbf{4}$matrices whose entries are all 1 , -1 , or 0 . 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.

Consider a 4x4 matrix A with rows $\stackrel{\mathbf{鈫�}}{{\mathbf{V}}_{\mathbf{1}}}\mathbf{,}\stackrel{\mathbf{鈫�}}{{\mathbf{V}}_{\mathbf{2}}}\mathbf{,}\stackrel{\mathbf{鈫�}}{{\mathbf{V}}_{\mathbf{3}}}\mathbf{,}\stackrel{\mathbf{鈫�}}{{\mathbf{V}}_{\mathbf{4}}}$. If det(A) = 8, find the determinants in Exercises 11 through 16.

16. role="math" localid="1659506283449" $\mathit{d}\mathit{e}\mathit{t}\mathbf{\left[}\begin{array}{c}\mathbf{6}{\stackrel{\mathbf{鈬赌}}{\mathbf{v}}}_{\mathbf{1}}\mathbf{+}\mathbf{2}{\stackrel{\mathbf{鈬赌}}{\mathbf{v}}}_{\mathbf{4}}\\ {\stackrel{\mathbf{鈬赌}}{\mathbf{v}}}_{\mathbf{2}}\\ {\stackrel{\mathbf{鈬赌}}{\mathbf{v}}}_{\mathbf{3}}\\ \mathbf{3}{\stackrel{\mathbf{鈬赌}}{\mathbf{v}}}_{\mathbf{1}}\mathbf{+}{\stackrel{\mathbf{鈬赌}}{\mathbf{v}}}_{\mathbf{4}}\end{array}\mathbf{\right]}$

Find the determinants of the linear transformations in Exercises 17 through 28.

17.$\mathit{T}\mathbf{\left(}\mathbf{f}\mathbf{\right)}\mathbf{=}\mathbf{2}\mathit{f}\mathbf{+}\mathbf{3}\mathit{f}\mathbf{\text{'}}\mathbf{}\mathbf{from}\mathbf{}{\mathit{P}}_{\mathbf{2}}\mathbf{}\mathbf{to}\mathbf{}{\mathit{P}}_{\mathbf{2}}$

25. If the determinant of a square matrix is -1, then Amust be an orthogonal matrix.

Find the area of the triangle defined by $\left[\begin{array}{c}3\\ 7\end{array}\right]$and$\left[\begin{array}{c}8\\ 2\end{array}\right]$.

IfA is a $\mathbf{4}\mathbf{脳}\mathbf{4}$ matrix whose entries are all 1 or -1 , then $\mathit{d}\mathit{e}\mathit{t}\mathit{A}$must be divisible by 8 (i.e., $\mathit{d}\mathit{e}\mathit{t}\mathit{A}\mathbf{=}\mathbf{8}\mathit{k}$for some integer k).

What is the area of the largest ellipse you can inscribe into a triangle with side lengths 3,4 , and 5 ? Hint: The largest ellipse you can inscribe into an equilateral triangle is a circle.

Which of the following functions F of $\mathbf{A}\mathbf{=}\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]$ are linear in both columns? Which are linear in both rows? Which are alternating on the columns?
role="math" localid="1659502796300" $$\begin{gathered} \mathit{a}\mathbf{.}\mathbf{}\mathit{F}\mathbf{\left(}\mathbf{A}\mathbf{\right)}\mathbf{=}\mathit{b}\mathit{c} \\ \mathit{b}\mathbf{.}\mathbf{}\mathit{F}\mathbf{\left(}\mathbf{A}\mathbf{\right)}\mathbf{=}\mathit{c}\mathit{d} \\ \mathit{c}\mathbf{.}\mathbf{}\mathit{F}\mathbf{\left(}\mathbf{A}\mathbf{\right)}\mathbf{=}\mathit{a}\mathit{c} \\ \mathit{d}\mathbf{.}\mathbf{}\mathit{F}\mathbf{\left(}\mathbf{A}\mathbf{\right)}\mathbf{=}\mathit{b}\mathit{c}\mathbf{-}\mathit{a}\mathit{d} \\ \mathit{e}\mathbf{.}\mathbf{}\mathit{F}\mathbf{\left(}\mathbf{A}\mathbf{\right)}\mathbf{=}\mathit{c} \end{gathered}$$