Precalculus Quiz: Using 2x2 Matrices For Plane Transformations

For the transformation matrix $R=\begin{pmatrix}0&-1\\1&0\end{pmatrix}$, which represents a rotation of the coordinate plane, what is the image of the point $(3,4)$ under $T(\mathbf{v})=R\mathbf{v}$? (This transformation has $\det(R)=1$, so areas are preserved.)

A triangle with vertices at $A(2, 1)$, $B(4, 1)$, and $C(2, 3)$ is transformed by the matrix $$ \begin{pmatrix} 1 & -1 \ 2 & 0 \end{pmatrix}

Consider the composition of two transformations: first a reflection across the line $y = x$ represented by matrix $R = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}$, followed by a scaling transformation $S = \begin{pmatrix} 3 & 0 \\ 0 & 2 \end{pmatrix}$. A circle with area $\pi$ square units undergoes this composition. What is the area of the final image?

For the transformation matrix $A=\begin{pmatrix}1&1\\1&1\end{pmatrix},$ the determinant indicates whether the transformation collapses area. What is $\det(A)$, and what does it imply about the area scaling factor $|\det(A)|$?​

Given the transformation matrix $A=\begin{pmatrix}-1&0\\0&1\end{pmatrix}$ applied to the coordinate plane by $T(\mathbf{v})=A\mathbf{v}$, what geometric transformation does $A$ represent? (Here $\det(A)=-1$, so orientation is reversed while area is preserved.)

Given the transformation matrix $A = \begin{pmatrix} -1 & 0 \\ 0 & 1 \end{pmatrix}$, which acts on points in the plane by $\begin{pmatrix} x' \\ y' \end{pmatrix} = A \begin{pmatrix} x \\ y \end{pmatrix}$, what is the image of $(4,-3)$?

First rotate the plane by $90^\circ$ counterclockwise using $R=\begin{pmatrix}0&-1\\1&0\end{pmatrix},$ then reflect across the $x$-axis using $F=\begin{pmatrix}1&0\\0&-1\end{pmatrix}.$ What is the combined transformation matrix (apply $R$ first, then $F$)?

A shear transformation is represented by the matrix $\begin{pmatrix} 1 & k \\ 0 & 1 \end{pmatrix}$ where $k$ is a real number. A student observes that when $k = 5$, a rectangle with area 8 square units is transformed to a parallelogram with area 8 square units. What can be concluded about the value of $k = -3$?

The transformation matrix $T = \begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix}$ is applied to a rectangle with area 12 square units. After applying transformation $T$ twice in succession, what is the area of the resulting figure?

First rotate points $90^\circ$ counterclockwise using $R=\begin{pmatrix}0&-1\\1&0\end{pmatrix},$ then reflect across the $x$-axis using $F=\begin{pmatrix}1&0\\0&-1\end{pmatrix}.$ What is the combined transformation matrix (apply $R$ first, then $F$)?

The transformation matrix $\begin{pmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{pmatrix}$ represents a rotation by angle $\theta$. A student claims that applying this transformation to a triangle will always preserve its area. Which statement best explains why this claim is correct?

For the transformation matrix $A=\begin{pmatrix}0&1\\1&0\end{pmatrix}$ applied to points by $T(\mathbf{v})=A\mathbf{v}$, what geometric transformation does $A$ represent? (Hint: $\det(A)=-1$ indicates orientation reversal.)

Given the transformation matrix $F=\begin{pmatrix}1&0\\0&-1\end{pmatrix}$ applied to vectors by $T(\mathbf{v})=F\mathbf{v}$, what is the result of applying $F$ to $\begin{pmatrix}2\\5\end{pmatrix}$? (Here $\det(F)=-1$, so area is preserved but orientation is reversed.)

A transformation matrix $T$ maps the parallelogram with vertices $(0,0)$, $(3,0)$, $(1,2)$, $(4,2)$ to a new parallelogram with vertices $(0,0)$, $(6,3)$, $(-2,4)$, $(4,7)$. What is the absolute value of the determinant of $T$?

Two transformation matrices $A = \begin{pmatrix} 2 & 1 \\ 0 & 3 \end{pmatrix}$ and $B = \begin{pmatrix} 1 & 0 \\ 2 & 1 \end{pmatrix}$ are applied to the same original figure. If the area of the figure after transformation $A$ is 30 square units, what is the area of the figure after transformation $B$?

A parallelogram in the coordinate plane has vertices that form vectors $\vec{u} = \begin{pmatrix} 3 \\ 1 \end{pmatrix}$ and $\vec{v} = \begin{pmatrix} 1 \\ 2 \end{pmatrix}$ as adjacent sides. If this parallelogram is transformed by matrix $$M = \begin{pmatrix} 2 & 1 \ 0 & 3 \end{pmatrix}

First rotate points $90^\circ$ counterclockwise using $R=\begin{pmatrix}0&-1\\1&0\end{pmatrix},$ then reflect across the $x$-axis using $F=\begin{pmatrix}1&0\\0&-1\end{pmatrix}.$ What is the combined transformation matrix (apply $R$ first, then $F$)?​

A transformation matrix has the property that it maps the unit square (with vertices at $(0,0)$, $(1,0)$, $(1,1)$, $(0,1)$) to a parallelogram with vertices at $(0,0)$, $(2,1)$, $(1,4)$, $(-1,3)$. What is the determinant of this transformation matrix?

For the transformation matrix $A=\begin{pmatrix}0&1\1&0\end{pmatrix}$, which swaps coordinates of each point, what is the determinant $\det(A)$ and what does its sign indicate about orientation?

Given the transformation matrix $S=\begin{pmatrix}3&0\\0&1\end{pmatrix}$ acting on the plane by $T(\mathbf{v})=S\mathbf{v}$, how does this transformation affect areas? (Use $|\det(S)|$ as the area scaling factor.)