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). For composition, applying R first then F gives the matrix FR = [1 0; 0 -1] [0 -1; 1 0] = [0 -1; -1 0], as calculated by matrix multiplication. Choice A is correct because it accurately computes the matrix product in the correct order. Choice C uses the wrong order for composition, computing RF when the correct order is FR. 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. 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).

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).

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 = [2 0; 0 -1], we calculate $\det(A) = 2*(-1) - 0*0 = -2 - 0 = -2$. This means the transformation scales all areas by a factor of $|-2| = 2$, so a region with area 1 would transform to area 2. Since $\det(A) = -2$ 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 correctly calculates the determinant and interprets its absolute value and sign. Choice D uses $\det(A)$ instead of $|\det(A)|$ for the area scaling factor, but area magnification is the absolute value of the determinant (areas can't be negative).

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.

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).

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 $A = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}$, we calculate $\det(A) = 00 - 11 = 0 - 1 = -1$. This means the transformation scales all areas by a factor of $|-1| = 1$, so a region with area 1 would transform to area 1. 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 B is correct because it correctly calculates the determinant. Choice D claims the transformation preserves orientation when $\det(A) = -1$, but positive determinant means orientation preserved and negative means reversed.

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.).

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 matrix A with det(A) = -2, the transformation scales all areas by a factor of |-2| = 2, so a region with area 3 would transform to area 6. Choice B is correct because it correctly identifies |det(A)| as the area scaling factor. Choice A uses det(A) instead of |det(A)| for the area scaling factor, but area magnification is the absolute value of the determinant (areas can't be negative). 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). 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).

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 S = [3 0; 0 1], we calculate det(S) = 31 - 00 = 3. This means the transformation scales all areas by a factor of |det(S)| = |3| = 3, so a region with area 1 would transform to area 3. Choice B is correct because it correctly identifies |det(S)| as the area scaling factor. Choice A uses det(S) instead of |det(S)| for the area scaling factor, but area magnification is the absolute value of the determinant (areas can't be negative). 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). 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.

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 [0 1; 1 0] has the form of a reflection because the first column [0; 1] and second column [1; 0] swap the standard basis vectors, reflecting across y = x. Choice B is correct because it properly describes the geometric transformation. Choice A incorrectly identifies the transformation as rotation 90° counterclockwise, but the matrix [0 1; 1 0] actually represents reflection across y = x because it swaps coordinates, and det = -1 indicates reflection. 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.