Determinants as Size Functions

Is

inside of the box formed by these three?

Find the volume of the region defined by the vectors.

• $\langle\begin{pmatrix}1\\ 3\end{pmatrix},\begin{pmatrix}-1\\ 4\end{pmatrix}\rangle$

• $\langle\begin{pmatrix}2\\ 1\\ 0\end{pmatrix},\begin{pmatrix}3\\ -2\\ 4\end{pmatrix},\begin{pmatrix}8\\ -3\\ 8\end{pmatrix}\rangle$

• $\langle\begin{pmatrix}1\\ 2\\ 0\\ 1\end{pmatrix},\begin{pmatrix}2\\ 2\\ 2\\ 2\end{pmatrix},\begin{pmatrix}-1\\ 3\\ 0\\ 5\end{pmatrix},\begin{pmatrix}0\\ 1\\ 0\\ 7\end{pmatrix}\rangle$

Suppose that $\mathchoice{\left|A\right|}{|A|}{|A|}{|A|}=3$. By what factor do these change volumes?

• (a)  $A$     (b)  $A^{2}$     (c)  $A^{-2}$

Consider the linear transformation $t$ of $\mathbb{R}^{3}$ represented with respect to the standard bases by this matrix.

• Compute the determinant of the matrix. Does the transformation preserve orientation or reverse it?

• Find the size of the box defined by these vectors. What is its orientation?

• Find the images under $t$ of the vectors in the prior item and find the size of the box that they define. What is…

By what factor does each transformation change the size of boxes?

• (a)  $\begin{pmatrix}x\\ y\end{pmatrix}\mapsto\begin{pmatrix}2x\\ 3y\end{pmatrix}$     (b)  $\begin{pmatrix}x\\ y\end{pmatrix}\mapsto\begin{pmatrix}3x-y\\ -2x+y\end{pmatrix}$     (c)  $\begin{pmatrix}x\\ y\\ z\end{pmatrix}\mapsto\begin{pmatrix}x-y\\ x+y+z\\ y-2z\end{pmatrix}$

What is the area of the image of the rectangle $[2..4]\times[2..5]$ under the action of this matrix?

If $t\colon\mathbb{R}^{3}\to\mathbb{R}^{3}$ changes volumes by a factor of $7$ and $s\colon\mathbb{R}^{3}\to\mathbb{R}^{3}$ changes volumes by a factor of $3/2$ then by what factor will their composition changes volumes?

In what way does the definition of a box differ from the definition of a span?

Does $\mathchoice{\left|TS\right|}{|TS|}{|TS|}{|TS|}=\mathchoice{\left|ST\right|}{|ST|}{|ST|}{|ST|}$? $\mathchoice{\left|T(SP)\right|}{|T(SP)|}{|T(SP)|}{|T(SP)|}=\mathchoice{\left|(TS)P\right|}{|(TS)P|}{|(TS)P|}{|(TS)P|}$?

Show that there are no $2\!\times\!2$ matrices $A$ and $B$ satisfying these.

• Suppose that $\mathchoice{\left|A\right|}{|A|}{|A|}{|A|}=3$ and that $\mathchoice{\left|B\right|}{|B|}{|B|}{|B|}=2$. Find $\mathchoice{\left|A^{2}\cdot{{B}^{\mathsf{T}}}\cdot B^{-2}\cdot{{A}^{\mathsf{T}}}\right|}{|A^{2}\cdot{{B}^{\mathsf{T}}}\cdot B^{-2}\cdot{{A}^{\mathsf{T}}}|}{|A^{2}\cdot{{B}^{\mathsf{T}}}\cdot B^{-2}\cdot{{A}^{\mathsf{T}}}|}{|A^{2}\cdot{{B}^{\mathsf{T}}}\cdot B^{-2}\cdot{{A}^{\mathsf{T}}}|}$.

• Assume that $\mathchoice{\left|A\right|}{|A|}{|A|}{|A|}=0$. Prove that $\mathchoice{\left|6A^{3}+5A^{2}+2A\right|}{|6A^{3}+5A^{2}+2A|}{|6A^{3}+5A^{2}+2A|}{|6A^{3}+5A^{2}+2A|}=0$.

Let $T$ be the matrix representing (with respect to the standard bases) the map that rotates plane vectors counterclockwise through $\theta$ radians. By what factor does $T$ change sizes?

Must a transformation $t\colon\mathbb{R}^{2}\to\mathbb{R}^{2}$ that preserves areas also preserve lengths?

What is the volume of a parallelepiped in $\mathbb{R}^{3}$ bounded by a linearly dependent set?

Find the area of the triangle in $\mathbb{R}^{3}$ with endpoints $(1,2,1)$, $(3,-1,4)$, and $(2,2,2)$. (This asks for area, not volume. The triangle defines a plane; what is the area of the triangle in that plane?)

An alternate proof of Theorem 1.5 uses the definition of determinant functions.

• Note that the vectors forming $S$ make a linearly dependent set if and only if $\mathchoice{\left|S\right|}{|S|}{|S|}{|S|}=0$, and check that the result holds in this case.

• For the $\mathchoice{\left|S\right|}{|S|}{|S|}{|S|}\neq 0$ case, to show that $\mathchoice{\left|TS\right|}{|TS|}{|TS|}{|TS|}/\mathchoice{\left|S\right|}{|S|}{|S|}{|S|}=\mathchoice{\left|T\right|}{|T|}{|T|}{|T|}$ for all transformations, consider the function $d\colon\mathcal{M}_{n\!\times\!n}\to\mathbb{R}$ given by $T\mapsto\mathchoice{\left|TS\right|}{|TS|}{|TS|}{|TS|}/\mathchoice{\left|S\right|}{|S|}{|S|}{|S|}$. Show that $d$ has the first property of a determinant.

• Show that $d$ has the remaining three propert…

Give a non-identity matrix with the property that ${{A}^{\mathsf{T}}}=A^{-1}$. Show that if ${{A}^{\mathsf{T}}}=A^{-1}$ then $\mathchoice{\left|A\right|}{|A|}{|A|}{|A|}=\pm 1$. Does the converse hold?

The algebraic property of determinants that factoring a scalar out of a single row will multiply the determinant by that scalar shows that where $H$ is $3\!\times\!3$, the determinant of $cH$ is $c^{3}$ times the determinant of $H$. Explain this geometrically, that is, using Theorem 1.5. (The observation that increasing the linear size of a three-dimensional object by a factor of $c$ will increase its volume by a factor of $c^{3}$…

We say that matrices $H$ and $G$ are similar if there is a nonsingular matrix $P$ such that $H=P^{-1}GP$ (we will study this relation in Chapter Five). Show that similar matrices have the same determinant.

This question uses material from the optional Determinant Functions Exist subsection. Prove Theorem 1.5 by using the permutation expansion formula for the determinant.