Using 2x2 Matrices for Plane Transformations - Precalculus

For the transformation matrix $F=\begin{pmatrix}1&0\\0&-1\end{pmatrix},$ which reflects points in the coordinate plane, what is the image of the vector $\begin{pmatrix}2\\5\end{pmatrix}?$
Answer: $\begin{pmatrix}2\\-5\end{pmatrix}$ This question tests understanding of how $2 \times 2$ matrices act as transformations of the coordinate plane. A $2 \times 2$ matrix A represents a transformation of the plane: when applied to a point or vector $[x; y]$ , it produces a new point/vector $[x'; y'] = A[x; y] = [ax+by; cx+dy]$ , effectively moving, rotating, reflecting, or scaling points according to the matrix entries. To transform vector $[2; 5]$ by matrix F = $[1 0; 0 -1]$ , we compute $[1 0; 0 -1][2; 5] = [1*2 + 0*5; 0*2 + (-1)*5] = [2; -5]$ , giving the image vector [2; -5]. Choice B is correct because it accurately computes the matrix-vector product. Choice C makes an error in the matrix-vector multiplication, calculating [-2; -5] perhaps by reflecting across both axes instead of just the x-axis. Common $2 \times 2$ transformations include rotations (like [0 -1; 1 0] for 90° counterclockwise), reflections (like [1 0; 0 -1] for reflection across the x-axis), and scalings (like [k 0; 0 k] for uniform scaling by factor k). Key to plane transformations: apply the matrix to a vector [x; y] by computing [ax+by; cx+dy], and remember that the determinant det(A) = ad - bc tells you about area scaling (|det| is the factor) and orientation (sign indicates preserved or reversed).
Given the transformation matrix $F=\begin{pmatrix}1&0\0&-1\end{pmatrix}$ applied to vectors in the plane, what geometric transformation does $F$ represent?
Answer: Reflection across the $x$ -axis This question tests understanding of how 2×2 matrices act as transformations of the coordinate plane. Common 2×2 transformations include rotations (like $\begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}$ for $90^\circ$ counterclockwise), reflections (like $\begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}$ for reflection across the x-axis), and scalings (like $\begin{pmatrix} k & 0 \\ 0 & k \end{pmatrix}$ for uniform scaling by factor k). The matrix $\begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}$ has the form of a reflection because the first column [1; 0] keeps the x-basis unchanged while the second column [0; -1] flips the y-basis, indicating reflection across the x-axis. Choice C is correct because it properly describes the geometric transformation. Choice A incorrectly identifies the transformation as reflection across the y-axis, but the matrix $\begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}$ actually represents reflection across the x-axis because it preserves x-coordinates and negates y-coordinates. To identify a transformation geometrically, see what it does to the standard basis vectors: [1; 0] maps to the first column of A, and [0; 1] maps to the second column, which reveals the transformation's nature (rotation, reflection, scaling, etc.). Common transformations to recognize: $\begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}$ rotates $90^\circ$ counterclockwise, $\begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}$ reflects across x-axis, $\begin{pmatrix} -1 & 0 \\ 0 & 1 \end{pmatrix}$ reflects across y-axis, $\begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}$ reflects across y = x, and $\begin{pmatrix} k & 0 \\ 0 & k \end{pmatrix}$ scales uniformly by k.
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?
Answer: $7$ The transformation matrix maps $(1,0) \to (2,1)$ and $(0,1) \to (-1,3)$ , so the matrix is $\begin{pmatrix} 2 & -1 \\ 1 & 3 \end{pmatrix}$ . The determinant is $2(3) - (-1)(1) = 6 + 1 = 7$ . We can verify: the parallelogram has adjacent sides $\vec{u} = (2,1)$ and $\vec{v} = (-1,3)$ , giving area $|2(3) - 1(-1)| = 7$ . Choice B has the wrong sign. Choices C and D result from incorrectly identifying the transformation vectors.
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 $F$ )?
Answer: $\begin{pmatrix}0&-1\\-1&0\end{pmatrix}$ This question tests understanding of how 2×2 matrices act as transformations of the coordinate plane. Common 2×2 transformations include rotations (like [0 -1; 1 0] for 90° counterclockwise), reflections (like [1 0; 0 -1] for reflection across the x-axis), and scalings (like [k 0; 0 k] for uniform scaling by factor k). The combined matrix for applying R first then F is F R = [[1 0]; [0 -1]] times [[0 -1]; [1 0]] = [[0 -1]; [-1 0]], as computed by matrix multiplication. Choice A is correct because it accurately computes the matrix product for the composition. Choice C uses the wrong order for composition, computing R F instead of F R. For composition of transformations, remember the order: 'apply A then B' is written B(Av) = (BA)v, with the matrix applied first written on the right—this is backward from reading left to right. To identify a transformation geometrically, see what it does to the standard basis vectors: [1; 0] maps to the first column of A, and [0; 1] maps to the second column, which reveals the transformation's nature (rotation, reflection, scaling, etc.).
A linear transformation maps the unit square to a parallelogram. If the transformation matrix is $A=\begin{pmatrix}1&2\\0&1\end{pmatrix},$ what is the area magnification factor $|\det(A)|$ ?
Answer: $1$ This question tests understanding of how $2\times2$ matrices act as transformations of the coordinate plane and how the determinant relates to area scaling. The absolute value of the determinant, $|\det(A)|$ , tells us by what factor the transformation multiplies areas: a unit square (area 1) maps to a parallelogram with area $|\det(A)|$ , and this ratio applies to all regions in the plane. For matrix $A = \begin{pmatrix} 1 & 2 \\ 0 & 1 \end{pmatrix}$ , we calculate $\det(A) = 1 \times 1 - 2 \times 0 = 1 - 0 = 1$ . This means the transformation scales all areas by a factor of $|\det(A)| = |1| = 1$ , so a region with area 5 would transform to area 5. Choice C is correct because it correctly identifies $|\det(A)|$ as the area scaling factor. Choice B confuses the determinant formula, computing something like the entry 2 instead of ad - bc. For area scaling, always use the absolute value: $|\det(A)|$ gives the area magnification factor, so if $\det(A) = -2$ , areas are scaled by 2 (the absolute value), and the negative sign indicates orientation reversal (a reflection component).
A transformation maps the unit square $[0,1]\times[0,1]$ to a parallelogram with area $2$ . For this transformation matrix $A$ , what must $|\det(A)|$ be?
Answer: $2$ This question tests understanding of how 2×2 matrices act as transformations of the coordinate plane and how the determinant relates to area scaling. The absolute value of the determinant, |det(A)|, tells us by what factor the transformation multiplies areas: a unit square (area 1) maps to a parallelogram with area |det(A)|, and this ratio applies to all regions in the plane. For a transformation that maps the unit square to a parallelogram with area 2, we have |det(A)| = 2. Choice A is correct because it correctly identifies |det(A)| as the area scaling factor. Choice D states that areas are scaled by det(A)² or √det(A) instead of |det(A)|. For area scaling, always use the absolute value: |det(A)| gives the area magnification factor, so if det(A) = -2, areas are scaled by 2 (the absolute value), and the negative sign indicates orientation reversal (a reflection component). The determinant of a 2×2 transformation matrix A represents the area scaling factor: if det(A) = k, then the transformation scales all areas by a factor of |k| (the absolute value), with positive determinant preserving orientation and negative determinant reversing it (indicating a reflection).
Given the transformation matrix $A=\begin{bmatrix}-1&0\\0&1\end{bmatrix},$ which statement correctly describes how the transformation affects areas and orientation in the plane?
Answer: Areas are scaled by factor $1$ and orientation is reversed This question tests understanding of how the determinant relates to area scaling and orientation. The absolute value of the determinant, |det(A)|, tells us by what factor the transformation multiplies areas: a unit square (area 1) maps to a parallelogram with area |det(A)|, and this ratio applies to all regions in the plane. For matrix A = [-1 0; 0 1], we calculate det(A) = ad - bc = (-1)(1) - (0)(0) = -1 - 0 = -1. This means the transformation scales all areas by a factor of |det(A)| = |-1| = 1, so areas are preserved. Since det(A) = -1 is negative, the transformation reverses orientation: if you traverse a region counterclockwise before transformation, you traverse clockwise after, indicating a reflection is involved. Choice C is correct because it accurately states that areas are scaled by factor 1 (preserved) and orientation is reversed (due to negative determinant). Choice B incorrectly claims orientation is preserved when det(A) = -1, but positive determinant means orientation preserved and negative means reversed. Key to plane transformations: apply the matrix to a vector [x; y] by computing [ax+by; cx+dy], and remember that the determinant det(A) = ad - bc tells you about area scaling (|det| is the factor) and orientation (sign indicates preserved or reversed).
For the transformation matrix $F=\begin{bmatrix}1&0\\0&-1\end{bmatrix}$ (reflection across the $x$ -axis), what is the result of applying $F$ to the vector $\begin{bmatrix}2\\5\end{bmatrix}?$
Answer: $\begin{bmatrix}2\\-5\end{bmatrix}$ This question tests understanding of how 2×2 matrices act as transformations of the coordinate plane. A 2×2 matrix A represents a transformation of the plane: when applied to a point or vector [x; y], it produces a new point/vector [x'; y'] = A[x; y] = [ax+by; cx+dy], effectively moving, rotating, reflecting, or scaling points according to the matrix entries. To transform vector [2; 5] by matrix F = [1 0; 0 -1], we compute [1 0; 0 -1][2; 5] = [1(2) + 0(5); 0(2) + (-1)(5)] = [2; -5], giving the image vector [2; -5]. Choice B is correct because it accurately computes the matrix-vector product as [2; -5], which reflects the y-coordinate across the x-axis while keeping the x-coordinate unchanged. Choice A incorrectly gives [-2; 5], which would be a reflection across the y-axis instead of the x-axis. Common transformations to recognize: [0 -1; 1 0] rotates 90° counterclockwise, [1 0; 0 -1] reflects across x-axis, [-1 0; 0 1] reflects across y-axis, [0 1; 1 0] reflects across y = x, and [k 0; 0 k] scales uniformly by k.
For the transformation matrix $A=\begin{pmatrix}3&0\\0&1\end{pmatrix},$ the determinant is related to how areas scale under the transformation. What is the area scaling factor $|\det(A)|$ for this transformation?
Answer: $3$ This question tests understanding of how 2×2 matrices act as transformations of the coordinate plane and how the determinant relates to area scaling. The determinant of a 2×2 transformation matrix A represents the area scaling factor: if det(A) = k, then the transformation scales all areas by a factor of |k| (the absolute value), with positive determinant preserving orientation and negative determinant reversing it (indicating a reflection). For matrix A = [3 0; 0 1], we calculate det(A) = 3*1 - 0*0 = 3 - 0 = 3. This means the transformation scales all areas by a factor of |det(A)| = |3| = 3, so a region with area 1 would transform to area 3. Choice B is correct because it correctly identifies |det(A)| as the area scaling factor. Choice C confuses the determinant formula, computing 3*3=9 perhaps by squaring the scaling factor instead of ad - bc. For area scaling, always use the absolute value: |det(A)| gives the area magnification factor, so if det(A) = -2, areas are scaled by 2 (the absolute value), and the negative sign indicates orientation reversal (a reflection component).
For the transformation matrix $S=\begin{pmatrix}2&0\\0&2\end{pmatrix},$ what geometric transformation does $S$ represent in the plane, and what is the area scaling factor $|\det(S)|$ ?
Answer: Uniform scaling by $2$ ; area factor $4$ This question tests understanding of how 2×2 matrices act as transformations of the coordinate plane and how the determinant relates to area scaling. Common 2×2 transformations include rotations (like [0 -1; 1 0] for 90° counterclockwise), reflections (like [1 0; 0 -1] for reflection across the x-axis), and scalings (like [k 0; 0 k] for uniform scaling by factor k). The matrix [2 0; 0 2] has the form of a uniform scaling because the diagonal entries are equal and off-diagonals are zero, scaling both axes by 2. For this matrix S, det(S) = 2*2 - 0*0 = 4, so |det(S)| = 4. Choice C is correct because it properly describes the geometric transformation and correctly identifies |det(S)| as the area scaling factor. Choice D describes the wrong geometric effect, claiming the transformation is a horizontal stretch by 2 when actually it scales uniformly by 2 in both directions. For area scaling, always use the absolute value: |det(A)| gives the area magnification factor, so if det(A) = -2, areas are scaled by 2 (the absolute value), and the negative sign indicates orientation reversal (a reflection component).
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 $(4,-3)$ ?
Answer: $(-4,-3)$ This question tests understanding of how $2 \times 2$ matrices act as transformations of the coordinate plane. A $2 \times 2$ matrix A represents a transformation of the plane: when applied to a point or vector $\begin{pmatrix} x \\ y \end{pmatrix}$ , it produces a new point/vector $\begin{pmatrix} x' \\ y' \end{pmatrix} = A \begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} ax + by \\ cx + dy \end{pmatrix}$ , effectively moving, rotating, reflecting, or scaling points according to the matrix entries. To transform point (4, -3) by matrix A = $\begin{pmatrix} -1 & 0 \\ 0 & 1 \end{pmatrix}$ , we compute $[-1 \times 4 + 0 \times (-3); 0 \times 4 + 1 \times (-3)] = [-4 + 0; 0 - 3] = [-4; -3]$ , giving the image point (-4, -3). Choice C is correct because it accurately computes the matrix-vector product. Choice A makes an error in the matrix-vector multiplication, calculating as if the matrix negated y instead of x. Key to plane transformations: apply the matrix to a vector $\begin{pmatrix} x \\ y \end{pmatrix}$ by computing $\begin{pmatrix} ax + by \\ cx + dy \end{pmatrix}$ , and remember that the determinant det(A) = ad - bc tells you about area scaling ( $|$ det $|$ is the factor) and orientation (sign indicates preserved or reversed). Common transformations to recognize: $\begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}$ rotates 90° counterclockwise, $\begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}$ reflects across x-axis, $\begin{pmatrix} -1 & 0 \\ 0 & 1 \end{pmatrix}$ reflects across y-axis, $\begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}$ reflects across y = x, and $\begin{pmatrix} k & 0 \\ 0 & k \end{pmatrix}$ scales uniformly by k.
When the matrix $A=\begin{pmatrix}3&0\\0&1\end{pmatrix}$ is applied to the coordinate plane, $|\det(A)|$ gives the area magnification factor. By what factor are areas scaled?
Answer: $3$ This question tests understanding of how $2 \times 2$ matrices act as transformations of the coordinate plane and how the determinant relates to area scaling. The determinant of a $2 \times 2$ transformation matrix A represents the area scaling factor: if $\det(A) = k$ , then the transformation scales all areas by a factor of $|k|$ (the absolute value), with positive determinant preserving orientation and negative determinant reversing it (indicating a reflection). For matrix $A = \begin{pmatrix} 3 & 0 \\ 0 & 1 \end{pmatrix}$ , we calculate $\det(A) = 3 \times 1 - 0 \times 0 = 3 - 0 = 3$ . This means the transformation scales all areas by a factor of $|\det(A)| = |3| = 3$ , so a region with area 1 would transform to area 3. Choice B is correct because it correctly identifies $|\det(A)|$ as the area scaling factor. Choice C confuses the determinant formula, computing $(3)^2 = 9$ instead of $ad - bc = 3$ . For area scaling, always use the absolute value: $|\det(A)|$ gives the area magnification factor, so if $\det(A) = -2$ , areas are scaled by 2 (the absolute value), and the negative sign indicates orientation reversal (a reflection component).
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.)
Answer: Reflection across the $y$ -axis This question tests understanding of how 2×2 matrices act as transformations of the coordinate plane and how the determinant relates to area scaling. Common 2×2 transformations include rotations (like [0 -1; 1 0] for 90° counterclockwise), reflections (like [1 0; 0 -1] for reflection across the x-axis), and scalings (like [k 0; 0 k] for uniform scaling by factor k). The matrix [-1 0; 0 1] has the form of a reflection because the first column [-1; 0] flips the x-basis, and second column [0; 1] keeps y, reflecting across the y-axis. Choice B is correct because it properly describes the geometric transformation. Choice A incorrectly identifies the transformation as reflection across x-axis, but the matrix [-1 0; 0 1] actually represents reflection across y-axis because it negates x while keeping y. To identify a transformation geometrically, see what it does to the standard basis vectors: [1; 0] maps to the first column of A, and [0; 1] maps to the second column, which reveals the transformation's nature (rotation, reflection, scaling, etc.). Common transformations to recognize: [0 -1; 1 0] rotates 90° counterclockwise, [1 0; 0 -1] reflects across x-axis, [-1 0; 0 1] reflects across y-axis, [0 1; 1 0] reflects across y = x, and [k 0; 0 k] scales uniformly by k.
When matrix $A=\begin{bmatrix}1&1\\2&2\end{bmatrix}$ is applied as a transformation of the plane, how does it affect areas?
Answer: It collapses regions to a line or point because $\det(A)=0$ This question tests understanding of how the determinant relates to area scaling. The determinant of a 2×2 transformation matrix A represents the area scaling factor: if det(A) = k, then the transformation scales all areas by a factor of |k| (the absolute value), with positive determinant preserving orientation and negative determinant reversing it (indicating a reflection). For matrix A = [1 1; 2 2], we calculate det(A) = ad - bc = (1)(2) - (1)(2) = 2 - 2 = 0. This means the transformation is singular (non-invertible) and collapses the entire plane onto a line, making all areas zero. Choice C is correct because it properly identifies that det(A) = 0 means the transformation collapses regions to a line or point. Choice A incorrectly claims areas are doubled because |det(A)| = 2, but det(A) = 0, not 2. For area scaling, always use the absolute value: |det(A)| gives the area magnification factor, but when det(A) = 0, the transformation is degenerate and collapses the plane to a lower dimension.
For the shear transformation matrix $A=\begin{pmatrix}1&2\\0&1\end{pmatrix}$ acting on vectors by $T(\mathbf{v})=A\mathbf{v}$ , what is the image of $\begin{pmatrix}1\\-1\end{pmatrix}$ ? (Note that $\det(A)=1$ , so areas are preserved.)
Answer: $\begin{pmatrix}-1\\-1\end{pmatrix}$ This question tests understanding of how 2×2 matrices act as transformations of the coordinate plane. A 2×2 matrix A represents a transformation of the plane: when applied to a point or vector [x; y], it produces a new point/vector [x'; y'] = A[x; y] = [ax+by; cx+dy], effectively moving, rotating, reflecting, or scaling points according to the matrix entries. To transform vector [1; -1] by matrix A = [1 2; 0 1], we compute [1 2; 0 1][1; -1] = [1*1 + 2*(-1); 0*1 + 1*(-1)] = [-1; -1], giving the image [-1; -1]. Choice A is correct because it accurately computes the matrix-vector product. Choice B makes an error in the matrix-vector multiplication, calculating perhaps without the shear or sign error, getting [1; 1] instead of [-1; -1]. Key to plane transformations: apply the matrix to a vector [x; y] by computing [ax+by; cx+dy], and remember that the determinant det(A) = ad - bc tells you about area scaling (|det| is the factor) and orientation (sign indicates preserved or reversed). To identify a transformation geometrically, see what it does to the standard basis vectors: [1; 0] maps to the first column of A, and [0; 1] maps to the second column, which reveals the transformation's nature (rotation, reflection, scaling, etc.).
Given that a plane transformation has $\det(A)=2$ , what is the determinant of the inverse transformation $A^{-1}$ (assuming $A$ is invertible)?
Answer: $\tfrac{1}{2}$ This question tests understanding of how the determinant relates to area scaling and inverse transformations. The determinant of a 2×2 transformation matrix A represents the area scaling factor: if det(A) = k, then the transformation scales all areas by a factor of |k| (the absolute value), with positive determinant preserving orientation and negative determinant reversing it (indicating a reflection). For inverse transformations, there's a fundamental relationship: det( $A^{-1}$ ) = 1/det(A). Since det(A) = 2, we have det( $A^{-1}$ ) = 1/2. Choice C is correct because it accurately calculates det( $A^{-1}$ ) = 1/det(A) = 1/2. Choice A incorrectly gives 2, perhaps thinking the inverse has the same determinant, when actually the determinants are reciprocals. Key to plane transformations: apply the matrix to a vector [x; y] by computing [ax+by; cx+dy], and remember that the determinant det(A) = ad - bc tells you about area scaling (|det| is the factor) and orientation (sign indicates preserved or reversed).
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.)
Answer: $\begin{pmatrix}2\\-5\end{pmatrix}$ This question tests understanding of how 2×2 matrices act as transformations of the coordinate plane and how the determinant relates to area scaling. A 2×2 matrix A represents a transformation of the plane: when applied to a point or vector [x; y], it produces a new point/vector [x'; y'] = A[x; y] = [ax+by; cx+dy], effectively moving, rotating, reflecting, or scaling points according to the matrix entries. To transform vector [2; 5] by matrix F = [1 0; 0 -1], we compute [1 0; 0 -1][2; 5] = [1*2 + 0*5; 0*2 + (-1)*5] = [2; -5], giving the image [2; -5]. Choice B is correct because it accurately computes the matrix-vector product. Choice A makes an error in the matrix-vector multiplication, calculating perhaps a reflection across y-axis or sign error on both, getting [-2; 5] instead of [2; -5]. Key to plane transformations: apply the matrix to a vector [x; y] by computing [ax+by; cx+dy], and remember that the determinant det(A) = ad - bc tells you about area scaling (|det| is the factor) and orientation (sign indicates preserved or reversed). For area scaling, always use the absolute value: |det(A)| gives the area magnification factor, so if det(A) = -2, areas are scaled by 2 (the absolute value), and the negative sign indicates orientation reversal (a reflection component).
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?
Answer: $6\pi$ square units The composition is $SR = \begin{pmatrix} 3 & 0 \\ 0 & 2 \end{pmatrix}\begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix} = \begin{pmatrix} 0 & 3 \\ 2 & 0 \end{pmatrix}$ . The determinant is $0(0) - 3(2) = -6$ , so the area scaling factor is $|-6| = 6$ . The final area is $\pi \times 6 = 6\pi$ . Choice A uses only $|\det(S)|/3$ . Choice B uses only $|\det(S)|/2$ . Choice C adds the individual determinants instead of computing the composition.
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?
Answer: 12 square units Applying transformation $T$ twice means computing $T^2 = \begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix}^2 = \begin{pmatrix} -1 & 0 \\ 0 & -1 \end{pmatrix}$ . The determinant of $T^2$ is $(-1)(-1) - (0)(0) = 1$ , so the area scaling factor is $|1| = 1$ . The area remains 12 square units. Choice A results from dividing by the square of $|\det(T)|$ . Choice B results from dividing by $|\det(T)|$ . Choice D results from multiplying by $|\det(T)|$ .
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 $k = -3$ ?
Answer: A rectangle with area 8 will transform to have area 8 square units For any shear matrix $\begin{pmatrix} 1 & k \\ 0 & 1 \end{pmatrix}$ , the determinant is $1(1) - k(0) = 1$ , regardless of the value of $k$ . Since $|\det| = 1$ for any $k$ , area is always preserved under shear transformations. The area remains 8 square units. Choice A incorrectly multiplies by $k$ . Choice B incorrectly divides by $|k|$ . Choice D incorrectly uses $|k|$ as a scaling factor.

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