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!

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 $\begin{pmatrix} x \\ y \end{pmatrix}$ and multiply by transformation matrix $T = \begin{pmatrix} a & b \\ c & d \end{pmatrix}$ using matrix multiplication: $T \begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} a & b \\ c & d \end{pmatrix} \begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} ax + by \\ cx + dy \end{pmatrix} = \begin{pmatrix} x' \\ y' \end{pmatrix}$ 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)$\cdot$(rotation matrix). The resulting product matrix represents the combined transformation in one step! The matrix $M = \begin{pmatrix} 2 & 0 \\ 0 & 1 \end{pmatrix}$ transforms $(x, y)$ to $(2x, y)$, doubling the x-coordinate while leaving y unchanged, which is a non-uniform scaling in the x-direction. Choice C correctly captures this effect by analyzing the diagonal entries, where 2 scales x and 1 keeps y the same. Distractors like choice A might reverse the coordinates affected, but remember the top-left entry scales x and bottom-right scales y in diagonal matrices. 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 $\begin{pmatrix} x \\ y \end{pmatrix}$. (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 building strong skills in non-uniform transformations—keep going!

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!

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! Scaling both coordinates by 3 means multiplying x and y by 3, represented by the diagonal matrix with 3 on both diagonals. Choice B correctly identifies [3 0; 0 3] as the uniform scaling matrix by factor 3. A distractor like choice A might result from confusing uniform scaling with non-uniform, where only one coordinate is scaled by 3. 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). Excellent work recognizing scaling matrices!

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 $\begin{bmatrix} x \\ y \end{bmatrix}$ and multiply by transformation matrix $T = \begin{bmatrix} a & b \\ c & d \end{bmatrix}$ using matrix multiplication: $T \begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} a & b \\ c & d \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} a x + b y \\ c x + d y \end{bmatrix} = \begin{bmatrix} x' \\ y' \end{bmatrix}$ where $(x', y')$ is the transformed point. Common transformation matrices include: ROTATION by angle $\theta$ counterclockwise = $\begin{bmatrix} \cos(\theta) & -\sin(\theta) \\ \sin(\theta) & \cos(\theta) \end{bmatrix}$ (example: 90° rotation uses $\theta=90°$ giving $\begin{bmatrix} 0 & -1 \\ 1 & 0 \end{bmatrix}$ since $\cos(90°)=0$ and $\sin(90°)=1$), REFLECTION across x-axis = $\begin{bmatrix} 1 & 0 \\ 0 & -1 \end{bmatrix}$ (keeps x same, negates y), REFLECTION across y-axis = $\begin{bmatrix} -1 & 0 \\ 0 & 1 \end{bmatrix}$ (negates x, keeps y same), SCALING by factor k = $\begin{bmatrix} k & 0 \\ 0 & k \end{bmatrix}$ (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! The matrix $A = \begin{bmatrix} 2 & 0 \\ 0 & 1 \end{bmatrix}$ transforms $(x, y)$ to $(2x, y)$, doubling the x-coordinate while leaving y unchanged. Choice B correctly describes this non-uniform scaling effect by analyzing the diagonal entries. A distractor like choice D might confuse it with uniform scaling, which would require 2 on both diagonals. Matrix multiplication for transformations: Given transformation matrix $\begin{bmatrix} a & b \\ c & d \end{bmatrix}$ and point $(x, y)$: (1) Write point as column vector $\begin{bmatrix} x \\ y \end{bmatrix}$. (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 $(x', y') = (a x + b y, c x + d y)$. Example: $\begin{bmatrix} 0 & -1 \\ 1 & 0 \end{bmatrix}$ 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° counterclockwise rotation! Identifying transformations from matrices: ROTATION matrices have form $\begin{bmatrix} \cos(\theta) & -\sin(\theta) \\ \sin(\theta) & \cos(\theta) \end{bmatrix}$—look for this pattern with cos and -sin in first row, sin and cos in second row. Common: $\begin{bmatrix} 0 & -1 \\ 1 & 0 \end{bmatrix}$ is 90° rotation, $\begin{bmatrix} -1 & 0 \\ 0 & -1 \end{bmatrix}$ is 180° rotation. REFLECTION matrices have form $\begin{bmatrix} \pm 1 & 0 \\ 0 & \pm 1 \end{bmatrix}$ with exactly one negative—$\begin{bmatrix} 1 & 0 \\ 0 & -1 \end{bmatrix}$ reflects across x-axis (y negated), $\begin{bmatrix} -1 & 0 \\ 0 & 1 \end{bmatrix}$ reflects across y-axis (x negated). SCALING matrices have equal diagonal entries $\begin{bmatrix} k & 0 \\ 0 & k \end{bmatrix}$—both coordinates multiplied by same k, or different entries $\begin{bmatrix} a & 0 \\ 0 & b \end{bmatrix}$ 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 building strong skills in matrix identification!

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 S(3,2) and T=[1 2; 0 1], S' = [13 + 22; 03 + 12] = [7; 2], so (7,2)—this is a shear transformation. Choice B correctly applies the matrix multiplication to get (7,2). Distractors like choice A might ignore the off-diagonal term, giving (3,8) from a misread. Even for non-standard matrices, follow the same multiplication steps and check calculations—keep going, you're improving!

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!

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). The matrix for reflection across the y-axis should negate x and keep y, which is [-1 0; 0 1]. Choice B correctly matches this form, representing the y-axis reflection. A distractor like choice A might swap it with x-axis reflection [1 0; 0 -1]. Use pattern recognition: look for diagonal ±1 with x negative for y-axis—test with (1,1) to confirm, and keep building your skills!