Geometry Quiz: Using 2x2 Matrices For Plane Transformations
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Using 2x2 Matrices For Plane TransformationsQuestion 1 of 20

A shear transformation is given by H=(1201)H=\begin{pmatrix}1&2\\0&1\end{pmatrix} . What is the image of the point (1,3)(1,3) under HH?

(7,3)(7,3)
(3,7)(3,7)
(1,5)(1,5)
(5,1)(5,1)
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Geometry Quiz: Using 2x2 Matrices For Plane Transformations

Practice Using 2x2 Matrices For Plane Transformations in Geometry with focused quiz questions that help you check what you know, review explanations, and build confidence with test-style prompts.

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This quiz focuses on Using 2x2 Matrices For Plane Transformations, giving you a quick way to practice the rules, question types, and explanations that matter most for Geometry.

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Question 1

A shear transformation is given by H=(1201)H=\begin{pmatrix}1&2\\0&1\end{pmatrix} . What is the image of the point (1,3)(1,3) under HH?

  1. (7,3)(7,3) (correct answer)
  2. (3,7)(3,7)
  3. (1,5)(1,5)
  4. (5,1)(5,1)
Explanation: This question tests your ability to use 2×22 \times 2 matrices to represent and perform plane transformations (rotations, reflections, scaling) by multiplying transformation matrices with point coordinate vectors. A 2×22 \times 2 matrix can represent a linear transformation of the plane: to transform a point (x,y)(x, y), write it as column vector [x;y][x; y] and multiply by transformation matrix T=(abcd)T = \begin{pmatrix} a & b \\ c & d \end{pmatrix} using matrix multiplication: T[x;y]=(abcd)[x;y]=[ax+by;cx+dy]=[x′;y′]T[x; y] = \begin{pmatrix} a & b \\ c & d \end{pmatrix} [x; y] = [a x + b y; c x + d y] = [x'; y'] where (x′,y′)(x', y') is the transformed point. Common transformation matrices include: ROTATION by angle θ\theta counterclockwise = (cos⁡(θ)−sin⁡(θ)sin⁡(θ)cos⁡(θ))\begin{pmatrix} \cos(\theta) & -\sin(\theta) \\ \sin(\theta) & \cos(\theta) \end{pmatrix} (example: 90° rotation uses θ=90°\theta=90° giving (0−110)\begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix} since cos⁡(90°)=0\cos(90°)=0 and sin⁡(90°)=1\sin(90°)=1), REFLECTION across x-axis = (100−1)\begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix} (keeps x same, negates y), REFLECTION across y-axis = (−1001)\begin{pmatrix} -1 & 0 \\ 0 & 1 \end{pmatrix} (negates x, keeps y same), SCALING by factor k = (k00k)\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! The shear matrix H=(1201)H = \begin{pmatrix} 1 & 2 \\ 0 & 1 \end{pmatrix} applied to (1,3)(1, 3) gives x′=1⋅1+2⋅3=7x' = 1 \cdot 1 + 2 \cdot 3 = 7, y′=0⋅1+1⋅3=3y' = 0 \cdot 1 + 1 \cdot 3 = 3, shifting x based on y while keeping y fixed, characteristic of a horizontal shear. Choice A correctly calculates this, adding twice the y-value to x for the shear effect. Distractors like choice B might swap coordinates or ignore the off-diagonal term, but remember off-diagonals introduce dependence between x and y in transformations like shears. Matrix multiplication for transformations: Given transformation matrix (abcd)\begin{pmatrix} a & b \\ c & d \end{pmatrix} and point (x,y)(x, y): (1) Write point as column vector [x;y][x; y]. (2) Multiply: first row of matrix times vector gives x'-coordinate = a⋅x+b⋅ya \cdot x + b \cdot y. Second row times vector gives y'-coordinate = c⋅x+d⋅yc \cdot x + d \cdot y. (3) Result is transformed point (x′,y′)=(ax+by,cx+dy)(x', y') = (a x + b y, c x + d y). Example: (0−110)\begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix} applied to (5,3)(5, 3): x′=0⋅5+(−1)⋅3=−3x' = 0 \cdot 5 + (-1) \cdot 3 = -3, y′=1⋅5+0⋅3=5y' = 1 \cdot 5 + 0 \cdot 3 = 5, so image is (−3,5)(-3, 5). That's a 90° counterclockwise rotation! Identifying transformations from matrices: ROTATION matrices have form (cos⁡(θ)−sin⁡(θ)sin⁡(θ)cos⁡(θ))\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: (0−110)\begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix} is 90° rotation, (−100−1)\begin{pmatrix} -1 & 0 \\ 0 & -1 \end{pmatrix} is 180° rotation. REFLECTION matrices have form (±100±1)\begin{pmatrix} \pm 1 & 0 \\ 0 & \pm 1 \end{pmatrix} with exactly one negative—(100−1)\begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix} reflects across x-axis (y negated), (−1001)\begin{pmatrix} -1 & 0 \\ 0 & 1 \end{pmatrix} reflects across y-axis (x negated). SCALING matrices have equal diagonal entries (k00k)\begin{pmatrix} k & 0 \\ 0 & k \end{pmatrix}—both coordinates multiplied by same k, or different entries (a00b)\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). Impressive work with shears—visualize how shapes distort to understand better!

Question 2

Which matrix represents a rotation of 180∘180^\circ about the origin?

  1. (0−110)\begin{pmatrix}0&-1\\1&0\end{pmatrix}
  2. (−100−1)\begin{pmatrix}-1&0\\0&-1\end{pmatrix} (correct answer)
  3. (100−1)\begin{pmatrix}1&0\\0&-1\end{pmatrix}
  4. (−1001)\begin{pmatrix}-1&0\\0&1\end{pmatrix}
Explanation: 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!

Question 3

Which matrix represents a reflection across the yy-axis?​

  1. (100−1)\begin{pmatrix}1&0\\0&-1\end{pmatrix}
  2. (−1001)\begin{pmatrix}-1&0\\0&1\end{pmatrix} (correct answer)
  3. (0−110)\begin{pmatrix}0&-1\\1&0\end{pmatrix}
  4. (1001)\begin{pmatrix}1&0\\0&1\end{pmatrix}
Explanation: 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. A reflection across the y-axis negates the x-coordinate while keeping the y-coordinate unchanged, transforming (x, y) to (-x, y). The matrix that performs this is [-1 0; 0 1]: when applied to [x; y], it gives [-1·x + 0·y; 0·x + 1·y] = [-x; y]. Choice B correctly shows this matrix [-1 0; 0 1]. Choice A [1 0; 0 -1] reflects across the x-axis (negates y), choice C [0 -1; 1 0] is a 90° rotation, and choice D [1 0; 0 1] is the identity matrix (no transformation). Identifying transformations from matrices: 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). The pattern of which coordinate gets negated tells you the reflection axis!

Question 4

Let T=[−1001].T=\begin{bmatrix}-1&0\\0&1\end{bmatrix}. Apply TT to the point (−5,2)(-5,2). What is the image point?

  1. (5,2)(5,2) (correct answer)
  2. (−5,−2)(-5,-2)
  3. (5,−2)(5,-2)
  4. (−5,2)(-5,2)
Explanation: 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)(x, y), write it as column vector [xy]\begin{bmatrix} x \\ y \end{bmatrix} and multiply by transformation matrix T=[abcd]T = \begin{bmatrix} a & b \\ c & d \end{bmatrix} using matrix multiplication: T[xy]=[abcd][xy]=[ax+bycx+dy]=[x′y′]T \begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} a & b \\ c & d \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} ax + by \\ cx + dy \end{bmatrix} = \begin{bmatrix} x' \\ y' \end{bmatrix} where (x′,y′)(x', y') is the transformed point. Common transformation matrices include: ROTATION by angle θθ counterclockwise = [cos⁡(θ)−sin⁡(θ)sin⁡(θ)cos⁡(θ)]\begin{bmatrix} \cos(θ) & -\sin(θ) \\ \sin(θ) & \cos(θ) \end{bmatrix} (example: 90° rotation uses θ=90°θ=90° giving [0−110]\begin{bmatrix} 0 & -1 \\ 1 & 0 \end{bmatrix} since cos⁡(90°)=0\cos(90°)=0 and sin⁡(90°)=1\sin(90°)=1), REFLECTION across x-axis = [100−1]\begin{bmatrix} 1 & 0 \\ 0 & -1 \end{bmatrix} (keeps x same, negates y), REFLECTION across y-axis = [−1001]\begin{bmatrix} -1 & 0 \\ 0 & 1 \end{bmatrix} (negates x, keeps y same), SCALING by factor k = [k00k]\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! Applying T=[−1001]T = \begin{bmatrix} -1 & 0 \\ 0 & 1 \end{bmatrix} to (−5,2)(-5, 2) gives [−1∗(−5)+0∗2;0∗(−5)+1∗2]=[52][-1*(-5) + 0*2; 0*(-5) + 1*2] = \begin{bmatrix} 5 \\ 2 \end{bmatrix}, so the image is (5,2)(5, 2). Choice A correctly computes the matrix multiplication, reflecting the point across the y-axis by negating the x-coordinate. A distractor like choice B might come from incorrectly negating both coordinates, confusing it with a 180° rotation. Matrix multiplication for transformations: Given transformation matrix [abcd]\begin{bmatrix} a & b \\ c & d \end{bmatrix} and point (x,y)(x, y): (1) Write point as column vector [xy]\begin{bmatrix} x \\ y \end{bmatrix}. (2) Multiply: first row of matrix times vector gives x'-coordinate = a⋅x+b⋅ya·x + b·y. Second row times vector gives y'-coordinate = c⋅x+d⋅yc·x + d·y. (3) Result is transformed point (x′,y′)=(ax+by,cx+dy)(x', y') = (ax+by, cx+dy). Example: [0−110]\begin{bmatrix} 0 & -1 \\ 1 & 0 \end{bmatrix} applied to (5,3)(5, 3): x' = 0⋅5+(−1)⋅3=−30·5 + (-1)·3 = -3, y' = 1⋅5+0⋅3=51·5 + 0·3 = 5, so image is (−3,5)(-3, 5). That's a 90° counterclockwise rotation! Identifying transformations from matrices: ROTATION matrices have form [cos⁡(θ)−sin⁡(θ)sin⁡(θ)cos⁡(θ)]\begin{bmatrix} \cos(θ) & -\sin(θ) \\ \sin(θ) & \cos(θ) \end{bmatrix}—look for this pattern with cos and -sin in first row, sin and cos in second row. Common: [0−110]\begin{bmatrix} 0 & -1 \\ 1 & 0 \end{bmatrix} is 90° rotation, [−100−1]\begin{bmatrix} -1 & 0 \\ 0 & -1 \end{bmatrix} is 180° rotation. REFLECTION matrices have form [±100±1]\begin{bmatrix} ±1 & 0 \\ 0 & ±1 \end{bmatrix} with exactly one negative—[100−1]\begin{bmatrix} 1 & 0 \\ 0 & -1 \end{bmatrix} reflects across x-axis (y negated), [−1001]\begin{bmatrix} -1 & 0 \\ 0 & 1 \end{bmatrix} reflects across y-axis (x negated). SCALING matrices have equal diagonal entries [k00k]\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}

Question 5

A point P(4,3)P(4,3) is transformed by the matrix R=(0−110).R=\begin{pmatrix}0&-1\\1&0\end{pmatrix}. Using column vectors, compute R(43)R\begin{pmatrix}4\\3\end{pmatrix} and give the image point P′(x′,y′)P'(x',y').

  1. (−3,4)(-3,4) (correct answer)
  2. (3,−4)(3,-4)
  3. (4,3)(4,3)
  4. (−4,−3)(-4,-3)
Explanation: 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! For this problem, apply R = [0 -1; 1 0] to [4; 3]: x' = 04 + (-1)3 = -3, y' = 14 + 03 = 4, so the image is (-3, 4). Choice A correctly applies the matrix multiplication to get (-3, 4), matching the 90° counterclockwise rotation that swaps and signs coordinates appropriately. A common distractor like choice B might result from mistakenly using a clockwise rotation or swapping signs incorrectly, but remember to follow the matrix entries precisely for accurate computation. 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). Keep practicing these calculations, and you'll master matrix transformations in no time!

Question 6

A triangle has vertices A(1,1)A(1,1), B(3,1)B(3,1), and C(2,3)C(2,3). The triangle is transformed by the matrix S=(2002)S=\begin{pmatrix}2&0\\0&2\end{pmatrix} . What are the coordinates of C′C' (the image of CC)?

  1. (4,6)(4,6) (correct answer)
  2. (2,6)(2,6)
  3. (6,4)(6,4)
  4. (4,3)(4,3)
Explanation: This question tests your ability to use 2×22 \times 2 matrices to represent and perform plane transformations (rotations, reflections, scaling) by multiplying transformation matrices with point coordinate vectors. A 2×22 \times 2 matrix can represent a linear transformation of the plane: to transform a point (x,y)(x, y), write it as column vector (xy)\begin{pmatrix} x \\ y \end{pmatrix} and multiply by transformation matrix T=(abcd)T = \begin{pmatrix} a & b \\ c & d \end{pmatrix} using matrix multiplication: T(xy)=(abcd)(xy)=(ax+bycx+dy)=(x′y′)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′)(x', y') is the transformed point. Common transformation matrices include: ROTATION by angle θ\theta counterclockwise = (cos⁡(θ)−sin⁡(θ)sin⁡(θ)cos⁡(θ))\begin{pmatrix} \cos(\theta) & -\sin(\theta) \\ \sin(\theta) & \cos(\theta) \end{pmatrix} (example: 90° rotation uses θ=90°\theta=90° giving (0−110)\begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix} since cos⁡(90°)=0\cos(90°)=0 and sin⁡(90°)=1\sin(90°)=1), REFLECTION across x-axis = (100−1)\begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix} (keeps x same, negates y), REFLECTION across y-axis = (−1001)\begin{pmatrix} -1 & 0 \\ 0 & 1 \end{pmatrix} (negates x, keeps y same), SCALING by factor k = (k00k)\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! Applying S=(2002)S = \begin{pmatrix} 2 & 0 \\ 0 & 2 \end{pmatrix} to C(2, 3) gives (4, 6), scaling the entire triangle uniformly by factor 2, enlarging distances from the origin while preserving shape. Choice A correctly multiplies both coordinates by 2, as the diagonal matrix applies the same factor to x and y. Distractors like choice B might scale only one coordinate or misread the point, but confirm by applying to all vertices: A(1,1) to (2,2), B(3,1) to (6,2), C(2,3) to (4,6). Matrix multiplication for transformations: Given transformation matrix (abcd)\begin{pmatrix} a & b \\ c & d \end{pmatrix} and point (x, y): (1) Write point as column vector (xy)\begin{pmatrix} x \\ y \end{pmatrix}

Question 7

Which matrix represents a 180∘180^\circ rotation about the origin?

  1. [0−110]\begin{bmatrix}0&-1\\1&0\end{bmatrix}
  2. [−100−1]\begin{bmatrix}-1&0\\0&-1\end{bmatrix} (correct answer)
  3. [100−1]\begin{bmatrix}1&0\\0&-1\end{bmatrix}
  4. [−1001]\begin{bmatrix}-1&0\\0&1\end{bmatrix}
Explanation: 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!

Question 8

Which transformation is represented by the matrix $$A=\begin{pmatrix} 1 & 0 \ 0 & -1 \end{pmatrix}

  1. Reflection across the yy-axis
  2. Reflection across the xx-axis (correct answer)
  3. Rotation 90∘90^\circ counterclockwise about the origin
  4. Uniform scaling by factor −1-1
Explanation: 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))\begin{pmatrix} \cos(\theta) & -\sin(\theta) \\ \sin(\theta) & \cos(\theta) \end{pmatrix} (example: 90° rotation uses θ=90° giving (0−110)\begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix} since cos(90°)=0 and sin(90°)=1), REFLECTION across x-axis = (100−1)\begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix} (keeps x same, negates y), REFLECTION across y-axis = (−1001)\begin{pmatrix} -1 & 0 \\ 0 & 1 \end{pmatrix} (negates x, keeps y same), SCALING by factor k = (k00k)\begin{pmatrix} k & 0 \\ 0 & k \end{pmatrix} (multiplies both coordinates by k, enlarges by factor k). The matrix A=[1 0; 0 -1] transforms (x,y) to (x, -y), which is exactly a reflection across the x-axis, as y is negated while x remains unchanged. Choice B correctly identifies this as reflection across the x-axis by recognizing the diagonal form with positive x and negative y scaling. A distractor like choice A might confuse it with y-axis reflection, which would have [-1 0; 0 1] instead. Remember the strategy: identify transformations by pattern—reflections have ±1±1 on diagonal with one negative; test with a point like (1,1) to see the effect, and you'll master this quickly!

Question 9

What plane transformation is represented by the matrix A=(100−1)?A=\begin{pmatrix}1 & 0\\ 0 & -1\end{pmatrix}?​

  1. Reflection across the yy-axis
  2. Rotation 90∘90^\circ counterclockwise about the origin
  3. Reflection across the xx-axis (correct answer)
  4. Uniform scaling by factor −1-1 (a 180∘180^\circ rotation) only
Explanation: 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. The matrix A = [1 0; 0 -1] transforms a point (x, y) to (1·x + 0·y, 0·x + (-1)·y) = (x, -y), which keeps the x-coordinate unchanged while negating the y-coordinate—this is exactly a reflection across the x-axis. Choice C correctly identifies this as reflection across the x-axis, where points above the axis flip below and vice versa. Choice A (reflection across y-axis) would have matrix [-1 0; 0 1], choice B (90° rotation) would have matrix [0 -1; 1 0], and choice D (180° rotation) would have matrix [-1 0; 0 -1]. Identifying transformations from matrices: 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). The pattern [1 0; 0 -1] preserves x-coordinates and negates y-coordinates, creating a mirror image across the x-axis!

Question 10

A point R(1,−4)R(1,-4) is transformed by M=(−1001).M=\begin{pmatrix}-1 & 0\\ 0 & 1\end{pmatrix}. What is the image point R′R'?

  1. (1,4)(1,4)
  2. (−1,−4)(-1,-4) (correct answer)
  3. (−1,4)(-1,4)
  4. (4,−1)(4,-1)
Explanation: This question tests your ability to use 2×22 \times 2 matrices to represent and perform plane transformations (rotations, reflections, scaling) by multiplying transformation matrices with point coordinate vectors. A 2×22 \times 2 matrix can represent a linear transformation of the plane: to transform a point (x,y)(x, y), write it as column vector [x;y][x; y] and multiply by transformation matrix T=[a b;c d]T = [a \, b; c \, d] using matrix multiplication: T[x;y]=[a b;c d][x;y]=[ax+by;cx+dy]=[x′;y′]T[x; y] = [a \, b; c \, d][x; y] = [ax+by; cx+dy] = [x'; y'] where (x′,y′)(x', y') is the transformed point. Common transformation matrices include: ROTATION by angle θ\theta counterclockwise = [cos⁡(θ) −sin⁡(θ);sin⁡(θ) cos⁡(θ)][\cos(\theta) \, -\sin(\theta); \sin(\theta) \, \cos(\theta)] (example: 90° rotation uses θ=90°\theta=90° giving [0 −1;1 0][0 \, -1; 1 \, 0] since cos⁡(90°)=0\cos(90°)=0 and sin⁡(90°)=1\sin(90°)=1), REFLECTION across x-axis = [1 0;0 −1][1 \, 0; 0 \, -1] (keeps x same, negates y), REFLECTION across y-axis = [−1 0;0 1][-1 \, 0; 0 \, 1] (negates x, keeps y same), SCALING by factor k = [k 0;0 k][k \, 0; 0 \, k] (multiplies both coordinates by k, enlarges by factor k). For R(1,-4) and M=[−1 0;0 1]M=[-1 \, 0; 0 \, 1], R′=[−1∗1+0∗(−4);0∗1+1∗(−4)]=[−1;−4]R' = [-1*1 + 0*(-4); 0*1 + 1*(-4)] = [-1; -4], so (-1,-4). Choice B correctly computes this y-axis reflection, negating x while keeping y. A distractor like choice A might omit the negation, giving (1,4). Always double-check the multiplication and visualize the reflection—you're progressing wonderfully!

Question 11

A transformation is a rotation 180∘180^\circ about the origin. Which matrix represents this rotation?

  1. (0−110)\begin{pmatrix}0 & -1\\ 1 & 0\end{pmatrix}
  2. (−100−1)\begin{pmatrix}-1 & 0\\ 0 & -1\end{pmatrix} (correct answer)
  3. (100−1)\begin{pmatrix}1 & 0\\ 0 & -1\end{pmatrix}
  4. (−1001)\begin{pmatrix}-1 & 0\\ 0 & 1\end{pmatrix}
Explanation: 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!

Question 12

Which matrix represents a reflection across the yy-axis?

  1. (100−1)\begin{pmatrix}1 & 0\\ 0 & -1\end{pmatrix}
  2. (−1001)\begin{pmatrix}-1 & 0\\ 0 & 1\end{pmatrix} (correct answer)
  3. (0−110)\begin{pmatrix}0 & -1\\ 1 & 0\end{pmatrix}
  4. (2002)\begin{pmatrix}2 & 0\\ 0 & 2\end{pmatrix}
Explanation: 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!

Question 13

What plane transformation is represented by the matrix A=(100−1)?A=\begin{pmatrix}1 & 0\\ 0 & -1\end{pmatrix}?

  1. Reflection across the xx-axis (correct answer)
  2. Reflection across the yy-axis
  3. Rotation 90∘90^\circ counterclockwise about the origin
  4. Uniform scaling by factor −1-1 (a 180∘180^\circ rotation)
Explanation: 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. The matrix A = [1 0; 0 -1] has the form of a reflection matrix with diagonal entries where one is positive and one is negative: when applied to point (x, y), it gives A[x; y] = [1·x + 0·y; 0·x + (-1)·y] = [x; -y], keeping x unchanged while negating y. Choice A correctly identifies this as a reflection across the x-axis, since points (x, y) map to (x, -y), which is exactly what happens when you flip a point across the horizontal axis. The other choices fail because: reflection across y-axis would negate x (giving [-x; y]), rotation 90° would swap and negate coordinates appropriately, and 180° rotation would negate both coordinates (giving [-x; -y]). Identifying transformations from matrices: 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). The pattern [1 0; 0 -1] preserves x-coordinates and negates y-coordinates, which geometrically means reflecting points across the x-axis!

Question 14

Let T=(−1001).T=\begin{pmatrix}-1 & 0\\ 0 & 1\end{pmatrix}. What is the image of point M(−7,2)M(-7,2) under TT?​

  1. (−7,−2)(-7,-2)
  2. (7,2)(7,2) (correct answer)
  3. (7,−2)(7,-2)
  4. (−7,2)(-7,2)
Explanation: 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. The matrix T = [-1 0; 0 1] represents a reflection across the y-axis (negates x, keeps y same). Applying this to point M(-7, 2): T[-7; 2] = [-1 0; 0 1][-7; 2] = [(-1)·(-7) + 0·2; 0·(-7) + 1·2] = [7 + 0; 0 + 2] = [7; 2], giving the image point (7, 2). Choice B correctly shows this result where x is negated (-7 → 7) and y stays the same (2 → 2). Choice A incorrectly keeps the negative x-coordinate, choice C negates y instead of keeping it the same, and choice D negates both coordinates. Identifying transformations from matrices: REFLECTION matrices have form [±1 0; 0 ±1] with exactly one negative—[-1 0; 0 1] reflects across y-axis (x negated), flipping points horizontally across the y-axis!

Question 15

A rotation of 180∘180^\circ about the origin is applied to point N(6,−1)N(6,-1). What is the image of NN?​

  1. (6,1)(6,1)
  2. (−6,−1)(-6,-1)
  3. (−6,1)(-6,1) (correct answer)
  4. (1,−6)(1,-6)
Explanation: 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. A 180° rotation about the origin has matrix R₁₈₀ = [-1 0; 0 -1], which negates both coordinates. Applying this to point N(6, -1): R₁₈₀[6; -1] = [-1 0; 0 -1][6; -1] = [(-1)·6 + 0·(-1); 0·6 + (-1)·(-1)] = [-6 + 0; 0 + 1] = [-6; 1], giving the image point (-6, 1). Choice C correctly shows this result where both coordinates are negated: x: 6 → -6 and y: -1 → 1. Choice A incorrectly keeps both coordinates positive, choice B negates y incorrectly to -1, and choice D swaps coordinates instead of negating. Identifying transformations from matrices: ROTATION matrices have form [cos(θ) -sin(θ); sin(θ) cos(θ)]—for 180° rotation, cos(180°) = -1 and sin(180°) = 0, giving [-1 0; 0 -1]. This rotates points halfway around the origin, ending up on the opposite side!

Question 16

Two transformation matrices A=(2101)A = \begin{pmatrix} 2 & 1 \\ 0 & 1 \end{pmatrix} and $$B = \begin{pmatrix} 1 & 0 \ -1 & 2 \end{pmatrix}

  1. 1414 units² because both matrices have the same determinant value (correct answer)
  2. 77 units² because matrix B has half the area scaling effect of matrix A
  3. 2828 units² because matrix B has twice the area scaling effect of matrix A
  4. 2121 units² because the area scaling effects have a 3:2 ratio favoring matrix B
Explanation: 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.

Question 17

Triangle ABCABC has vertices A(1,1)A(1,1), B(3,1)B(3,1), and C(2,4)C(2,4). It is transformed by the matrix K=(2002).K=\begin{pmatrix}2 & 0\\ 0 & 2\end{pmatrix}. What are the coordinates of C′C' (the image of CC)?​

  1. (4,4)(4,4)
  2. (4,8)(4,8) (correct answer)
  3. (2,8)(2,8)
  4. (1,2)(1,2)
Explanation: 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. The matrix K = [2 0; 0 2] represents uniform scaling by factor 2. Applying this to vertex C(2, 4): K[2; 4] = [2 0; 0 2][2; 4] = [2·2 + 0·4; 0·2 + 2·4] = [4 + 0; 0 + 8] = [4; 8], giving C'(4, 8). Choice B correctly shows this result where both coordinates are doubled: x: 2 → 4 and y: 4 → 8. Choice A incorrectly has y' = 4 (not doubled), choice C incorrectly has x' = 2 (not doubled), and choice D shows the original point divided by 2 instead of multiplied. Matrix multiplication for transformations: SCALING matrices have equal diagonal entries [k 0; 0 k]—both coordinates multiplied by same k. The scaling transformation K = [2 0; 0 2] doubles all distances from the origin, enlarging the triangle by factor 2!

Question 18

Triangle △PQR\triangle PQR has vertices P(1,1)P(1,1), Q(3,1)Q(3,1), and R(2,3)R(2,3). A transformation is given by T=(2002).T=\begin{pmatrix}2 & 0\\ 0 & 2\end{pmatrix}. What are the coordinates of the image triangle △P′Q′R′\triangle P'Q'R'?

  1. P′(2,2), Q′(6,2), R′(4,6)P'(2,2),\ Q'(6,2),\ R'(4,6) (correct answer)
  2. P′(2,2), Q′(5,2), R′(4,5)P'(2,2),\ Q'(5,2),\ R'(4,5)
  3. P′(12,12), Q′(32,12), R′(1,32)P'(\tfrac{1}{2},\tfrac{1}{2}),\ Q'(\tfrac{3}{2},\tfrac{1}{2}),\ R'(1,\tfrac{3}{2})
  4. P′(−2,−2), Q′(−6,−2), R′(−4,−6)P'(-2,-2),\ Q'(-6,-2),\ R'(-4,-6)
Explanation: 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. The matrix T = [2 0; 0 2] uniformly scales by factor 2, so we apply it to each vertex: P'= T[1; 1] = [2·1; 2·1] = [2; 2], Q'= T[3; 1] = [2·3; 2·1] = [6; 2], R'= T[2; 3] = [2·2; 2·3] = [4; 6]. Choice A correctly shows P'(2,2), Q'(6,2), R'(4,6) as the scaled triangle with all distances from origin doubled. The transformation preserves the triangle's shape while doubling its size—each coordinate is multiplied by 2, making the image triangle similar to the original with scale factor 2. Matrix multiplication for transformations: Uniform scaling [2 0; 0 2] multiplies both coordinates by 2. For triangle vertices: P(1,1)→P'(2,2), Q(3,1)→Q'(6,2), R(2,3)→R'(4,6). The scaling preserves angles and ratios of distances, creating a similar triangle that's twice as large!

Question 19

A transformation matrix T=(2103)T = \begin{pmatrix} 2 & 1 \\ 0 & 3 \end{pmatrix} is applied to a triangle with vertices at (0,0)(0,0), (2,0)(2,0), and (0,1)(0,1). What is the ratio of the area of the transformed triangle to the original triangle?

  1. 3:13:1
  2. 5:15:1
  3. 6:16:1 (correct answer)
  4. 7:17:1
Explanation: 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.

Question 20

Triangle ABCABC has vertices A(1,1)A(1,1), B(3,1)B(3,1), and C(2,3)C(2,3) on the coordinate plane. The triangle is transformed by S=(2002)S = \begin{pmatrix} 2 & 0 \\ 0 & 2 \end{pmatrix}. Which set of vertices gives the image triangle A′B′C′A'B'C'?

  1. A′(2,2),B′(6,2),C′(4,6)A'(2,2), B'(6,2), C'(4,6) (correct answer)
  2. A′(2,2),B′(5,2),C′(4,5)A'(2,2), B'(5,2), C'(4,5)
  3. A′(12,12),B′(32,12),C′(1,32)A'(\tfrac{1}{2},\tfrac{1}{2}), B'(\tfrac{3}{2},\tfrac{1}{2}), C'(1,\tfrac{3}{2})
  4. A′(−2,−2),B′(−6,−2),C′(−4,−6)A'(-2,-2), B'(-6,-2), C'(-4,-6)
Explanation: 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][x; y] and multiply by transformation matrix T=[a b;c d]T = [a\ b; c\ d] using matrix multiplication: T[x;y]=[a b;c d][x;y]=[ax+by;cx+dy]=[x′;y′]T[x; y] = [a\ b; c\ d][x; y] = [ax + by; cx + dy] = [x'; y'] where (x′,y′)(x', y') is the transformed point. Common transformation matrices include: ROTATION by angle θ counterclockwise = [cos⁡(θ)−sin⁡(θ);sin⁡(θ) cos⁡(θ)][\cos(\theta) -\sin(\theta); \sin(\theta)\ \cos(\theta)] (example: 90° rotation uses θ=90° giving [0 −1;1 0][0\ -1; 1\ 0] since cos⁡(90°)=0\cos(90°)=0 and sin⁡(90°)=1\sin(90°)=1), REFLECTION across x-axis = [1 0;0 −1][1\ 0; 0\ -1] (keeps x same, negates y), REFLECTION across y-axis = [−1 0;0 1][-1\ 0; 0\ 1] (negates x, keeps y same), SCALING by factor k = [k 0;0 k][k\ 0; 0\ k] (multiplies both coordinates by k, enlarges by factor k). Applying S=[2 0;0 2]S=[2\ 0; 0\ 2] to each vertex: A(1,1)→(2,2), B(3,1)→(6,2), C(2,3)→(4,6). Choice A correctly scales all vertices by factor 2. Distractors like choice C might use scaling by 1/2 instead, shrinking the triangle. Transform each point individually and plot to verify the shape enlarges proportionally—great job, you're getting stronger!