This question tests your ability to use 2×2 matrices to represent and perform plane transformations (rotations, reflections, scaling) by multiplying transformation matrices with point coordinate vectors. A 2×2 matrix can represent a linear transformation of the plane: to transform a point (x, y), write it as column vector $[x; y]$ and multiply by transformation matrix $T = \begin{pmatrix} a & b \\ c & d \end{pmatrix}$ using matrix multiplication: $T[x; y] = \begin{pmatrix} a & b \\ c & d \end{pmatrix}[x; y] = [ax+by; cx+dy] = [x'; y']$ where (x', y') is the transformed point. Common transformation matrices include: ROTATION by angle θ counterclockwise = $[\cos(θ) -\sin(θ); \sin(θ) \cos(θ)]$ (example: 90° rotation uses θ=90° giving $[0 -1; 1 0]$ since cos(90°)=0 and sin(90°)=1), REFLECTION across x-axis = $[1 0; 0 -1]$ (keeps x same, negates y), REFLECTION across y-axis = $[-1 0; 0 1]$ (negates x, keeps y same), SCALING by factor k = $[k 0; 0 k]$ (multiplies both coordinates by k, enlarges by factor k). For Q(-2,5) and S=[3 0; 0 3], Q' = [3*(-2) + 05; 0(-2) + 3*5] = [-6; 15], so (-6,15). Choice A correctly computes this uniform scaling by factor 3, enlarging both coordinates proportionally. Distractors like choice B might forget to scale the y-coordinate fully, resulting in (-6,8) from a miscalculation. Practice by applying the matrix to simple points and verifying distances scale by k— you're doing great and will get even better!

This question tests your ability to use 2×2 matrices to represent and perform plane transformations (rotations, reflections, scaling) by multiplying transformation matrices with point coordinate vectors. A 2×2 matrix can represent a linear transformation of the plane: to transform a point (x, y), write it as column vector [x; y] and multiply by transformation matrix T = [a b; c d] using matrix multiplication: T[x; y] = [a b; c d][x; y] = [ax+by; cx+dy] = [x'; y'] where (x', y') is the transformed point. Common transformation matrices include: ROTATION by angle θ counterclockwise = [cos(θ) -sin(θ); sin(θ) cos(θ)] (example: 90° rotation uses θ=90° giving [0 -1; 1 0] since cos(90°)=0 and sin(90°)=1), REFLECTION across x-axis = [1 0; 0 -1] (keeps x same, negates y), REFLECTION across y-axis = [-1 0; 0 1] (negates x, keeps y same), SCALING by factor k = [k 0; 0 k] (multiplies both coordinates by k, enlarges by factor k). COMPOSITIONS of transformations: multiply matrices in reverse order (rightmost applied first)—to rotate then scale, compute (scaling matrix)·(rotation matrix). The resulting product matrix represents the combined transformation in one step! A 180° rotation matrix is [-1 0; 0 -1], as it matches [cos(180°) -sin(180°); sin(180°) cos(180°)] = [-1 0; 0 -1], negating both coordinates to flip the point through the origin. Choice B correctly identifies this matrix by recognizing the equal negative diagonals and zero off-diagonals, distinct from reflections or 90° rotations. Distractors like choice A might confuse with 90° rotation, but note the off-diagonal signs and values differ—use the general rotation formula to verify angles. Matrix multiplication for transformations: Given transformation matrix [a b; c d] and point (x, y): (1) Write point as column vector [x; y]. (2) Multiply: first row of matrix times vector gives x'-coordinate = a·x + b·y. Second row times vector gives y'-coordinate = c·x + d·y. (3) Result is transformed point (x', y') = (ax+by, cx+dy). Example: [0 -1; 1 0] applied to (5, 3): x' = 0·5 + (-1)·3 = -3, y' = 1·5 + 0·3 = 5, so image is (-3, 5). That's a 90° counterclockwise rotation! Identifying transformations from matrices: ROTATION matrices have form [cos(θ) -sin(θ); sin(θ) cos(θ)]—look for this pattern with cos and -sin in first row, sin and cos in second row. Common: [0 -1; 1 0] is 90° rotation, [-1 0; 0 -1] is 180° rotation. REFLECTION matrices have form [±1 0; 0 ±1] with exactly one negative—[1 0; 0 -1] reflects across x-axis (y negated), [-1 0; 0 1] reflects across y-axis (x negated). SCALING matrices have equal diagonal entries [k 0; 0 k]—both coordinates multiplied by same k, or different entries [a 0; 0 b] for non-uniform scaling. Pattern recognition allows identification without calculation! Checking: after transforming point, verify result makes sense geometrically (rotation should preserve distance from origin, reflection should mirror across axis, scaling should change distances proportionally). Wonderful progress on rotations—experiment with different angles!

This question tests your ability to use 2×2 matrices to represent and perform plane transformations (rotations, reflections, scaling) by multiplying transformation matrices with point coordinate vectors. A 2×2 matrix can represent a linear transformation of the plane: to transform a point (x, y), write it as column vector [x; y] and multiply by transformation matrix T = [a b; c d] using matrix multiplication: T[x; y] = [a b; c d][x; y] = [ax+by; cx+dy] = [x'; y'] where (x', y') is the transformed point. Common transformation matrices include: ROTATION by angle θ counterclockwise = [cos(θ) -sin(θ); sin(θ) cos(θ)] (example: 90° rotation uses θ=90° giving [0 -1; 1 0] since cos(90°)=0 and sin(90°)=1), REFLECTION across x-axis = [1 0; 0 -1] (keeps x same, negates y), REFLECTION across y-axis = [-1 0; 0 1] (negates x, keeps y same), SCALING by factor k = [k 0; 0 k] (multiplies both coordinates by k, enlarges by factor k). COMPOSITIONS of transformations: multiply matrices in reverse order (rightmost applied first)—to rotate then scale, compute (scaling matrix)·(rotation matrix). The resulting product matrix represents the combined transformation in one step! A 180° rotation transforms (x, y) to (-x, -y), matching the matrix [-1 0; 0 -1] from the rotation formula with θ=180°. Choice B correctly identifies this matrix by recognizing the pattern of -1 on both diagonals. A distractor like choice A might confuse it with 90° rotation, which has off-diagonal entries. Matrix multiplication for transformations: Given transformation matrix [a b; c d] and point (x, y): (1) Write point as column vector [x; y]. (2) Multiply: first row of matrix times vector gives x'-coordinate = a·x + b·y. Second row times vector gives y'-coordinate = c·x + d·y. (3) Result is transformed point (x', y') = (ax+by, cx+dy). Example: [0 -1; 1 0] applied to (5, 3): x' = 0·5 + (-1)·3 = -3, y' = 1·5 + 0·3 = 5, so image is (-3, 5). That's a 90° counterclockwise rotation! Identifying transformations from matrices: ROTATION matrices have form [cos(θ) -sin(θ); sin(θ) cos(θ)]—look for this pattern with cos and -sin in first row, sin and cos in second row. Common: [0 -1; 1 0] is 90° rotation, [-1 0; 0 -1] is 180° rotation. REFLECTION matrices have form [±1 0; 0 ±1] with exactly one negative—[1 0; 0 -1] reflects across x-axis (y negated), [-1 0; 0 1] reflects across y-axis (x negated). SCALING matrices have equal diagonal entries [k 0; 0 k]—both coordinates multiplied by same k, or different entries [a 0; 0 b] for non-uniform scaling. Pattern recognition allows identification without calculation! Checking: after transforming point, verify result makes sense geometrically (rotation should preserve distance from origin, reflection should mirror across axis, scaling should change distances proportionally). Well done spotting the rotation matrix!

det(A) = 2×1 - 1×0 = 2, and det(B) = 1×2 - 0×(-1) = 2. Both have the same area scaling factor of 2, so they produce equal areas when applied to the same original region. Choices B, C, and D incorrectly calculate the determinants or their relationship.

This question tests your ability to use 2×2 matrices to represent and perform plane transformations (rotations, reflections, scaling) by multiplying transformation matrices with point coordinate vectors. A 2×2 matrix can represent a linear transformation of the plane: to transform a point (x, y), write it as column vector [x; y] and multiply by transformation matrix T = [a b; c d] using matrix multiplication: T[x; y] = [a b; c d][x; y] = [ax+by; cx+dy] = [x'; y'] where (x', y') is the transformed point. Common transformation matrices include: ROTATION by angle θ counterclockwise = [cos(θ) -sin(θ); sin(θ) cos(θ)] (example: 90° rotation uses θ=90° giving [0 -1; 1 0] since cos(90°)=0 and sin(90°)=1), REFLECTION across x-axis = [1 0; 0 -1] (keeps x same, negates y), REFLECTION across y-axis = [-1 0; 0 1] (negates x, keeps y same), SCALING by factor k = [k 0; 0 k] (multiplies both coordinates by k, enlarges by factor k). For 180° rotation, the matrix is [cos(180) -sin(180); sin(180) cos(180)] = [-1 0; 0 -1], transforming (x,y) to (-x,-y). Choice B correctly identifies this matrix as the 180° rotation. Distractors like choice A might confuse it with 90° rotation [0 -1; 1 0]. Recall the rotation formula and match the trig values—practice with angles like 0°, 90°, 180°, and you'll ace these!

The ratio of areas under a linear transformation equals the absolute value of the determinant of the transformation matrix. For matrix T, det(T) = (2)(3) - (1)(0) = 6. Therefore, the area ratio is 6:1. Choice A uses only one diagonal element, choice B adds the matrix elements, and choice D uses the sum of all elements.

This question tests your ability to use 2×2 matrices to represent and perform plane transformations (rotations, reflections, scaling) by multiplying transformation matrices with point coordinate vectors. A 2×2 matrix can represent a linear transformation of the plane: to transform a point (x, y), write it as column vector [x; y] and multiply by transformation matrix T = [a b; c d] using matrix multiplication: T[x; y] = [a b; c d][x; y] = [ax+by; cx+dy] = [x'; y'] where (x', y') is the transformed point. For matrix M = [0 1; 1 0], applying to point (x, y): M[x; y] = [0 1; 1 0][x; y] = [0·x + 1·y; 1·x + 0·y] = [y; x], which swaps the x and y coordinates—this is exactly reflection across the line y = x. Choice A correctly identifies this as reflection across the line y = x, where each point (x, y) maps to (y, x). Choice B (reflection across x-axis) would need [1 0; 0 -1], C (90° rotation) would need [0 -1; 1 0], and D (scaling by 2) would need [2 0; 0 2]. The matrix [0 1; 1 0] has 1's on the anti-diagonal and 0's on the main diagonal, creating the coordinate swap (x, y) → (y, x). This reflection across y = x mirrors points across the 45° line through the origin where x equals y!

Rotation matrices have determinant 1 (preserve area), and the scaling matrix has determinant 4. Combined determinant is 1×4=4, so original area = 32/4 = 8. Choice A incorrectly claims rotation doubles area, choice C confuses linear and area scaling, choice D ignores the scaling effect.

Shear transformations have determinant 1, so they preserve area regardless of the shear parameter k. The original area 4×3=12 remains unchanged. The vertex displacement helps find k=3, but doesn't affect area preservation. Choices B, C, and D incorrectly assume shear transformations change area.

This question tests your ability to use $2 \times 2$ matrices to represent and perform plane transformations (rotations, reflections, scaling) by multiplying transformation matrices with point coordinate vectors. A $2 \times 2$ matrix can represent a linear transformation of the plane: to transform a point $(x, y)$, write it as column vector $[x; y]$ and multiply by transformation matrix $T = \begin{pmatrix} a & b \\ c & d \end{pmatrix}$ using matrix multiplication: $T[x; y] = \begin{pmatrix} a & b \\ c & d \end{pmatrix} [x; y] = [ax+by; cx+dy] = [x'; y']$ where $(x', y')$ is the transformed point. Common transformation matrices include: ROTATION by angle $\theta$ counterclockwise = $\begin{pmatrix} \cos(\theta) & -\sin(\theta) \\ \sin(\theta) & \cos(\theta) \end{pmatrix}$ (example: $90^\circ$ rotation uses $\theta=90^\circ$ giving $\begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}$ since $\cos(90^\circ)=0$ and $\sin(90^\circ)=1$), REFLECTION across x-axis = $\begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}$ (keeps x same, negates y), REFLECTION across y-axis = $\begin{pmatrix} -1 & 0 \\ 0 & 1 \end{pmatrix}$ (negates x, keeps y same), SCALING by factor k = $\begin{pmatrix} k & 0 \\ 0 & k \end{pmatrix}$ (multiplies both coordinates by k, enlarges by factor k). COMPOSITIONS of transformations: multiply matrices in reverse order (rightmost applied first)—to rotate then scale, compute (scaling matrix)·(rotation matrix). The resulting product matrix represents the combined transformation in one step! Here, $T = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}$ applied to a general $(x, y)$ gives $(x, -y)$, which keeps x the same and negates y, matching a reflection across the x-axis. Choice C correctly identifies this as reflection across the x-axis based on the matrix entries where the y-component is negated while x remains unchanged. A distractor like choice A might confuse it with y-axis reflection, which would have $\begin{pmatrix} -1 & 0 \\ 0 & 1 \end{pmatrix}$ instead, so always check which diagonal entry is negative to distinguish axis reflections. Matrix multiplication for transformations: Given transformation matrix $\begin{pmatrix} a & b \\ c & d \end{pmatrix}$ and point $(x, y)$: (1) Write point as column vector $[x; y]$. (2) Multiply: first row of matrix times vector gives x'-coordinate = $a \cdot x + b \cdot y$. Second row times vector gives y'-coordinate = $c \cdot x + d \cdot y$. (3) Result is transformed point $(x', y') = (ax+by, cx+dy)$. Example: $\begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}$ applied to $(5, 3)$: x' = $0 \cdot 5 + (-1) \cdot 3 = -3$, y' = $1 \cdot 5 + 0 \cdot 3 = 5$, so image is $(-3, 5)$. That's a $90^\circ$ counterclockwise rotation! Identifying transformations from matrices: ROTATION matrices have form $\begin{pmatrix} \cos(\theta) & -\sin(\theta) \\ \sin(\theta) & \cos(\theta) \end{pmatrix}$—look for this pattern with cos and -sin in first row, sin and cos in second row. Common: $\begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}$ is $90^\circ$ rotation, $\begin{pmatrix} -1 & 0 \\ 0 & -1 \end{pmatrix}$ is $180^\circ$ rotation. REFLECTION matrices have form $\begin{pmatrix} \pm 1 & 0 \\ 0 & \pm 1 \end{pmatrix}$ with exactly one negative—$\begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}$ reflects across x-axis (y negated), $\begin{pmatrix} -1 & 0 \\ 0 & 1 \end{pmatrix}$ reflects across y-axis (x negated). SCALING matrices have equal diagonal entries $\begin{pmatrix} k & 0 \\ 0 & k \end{pmatrix}$—both coordinates multiplied by same k, or different entries $\begin{pmatrix} a & 0 \\ 0 & b \end{pmatrix}$ for non-uniform scaling. Pattern recognition allows identification without calculation! Checking: after transforming point, verify result makes sense geometrically (rotation should preserve distance from origin, reflection should mirror across axis, scaling should change distances proportionally). You're doing great—keep matching matrix patterns to transformation types!