Using 2x2 Matrices for Plane Transformations - Geometry

A transformation is defined by $A = \begin{bmatrix} 2 & 0 \\ 0 & 1 \end{bmatrix}.$ What does this transformation do to a point $(x, y)$ ?
Answer: Doubles the $x$ -coordinate and leaves the $y$ -coordinate unchanged 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!
A point $S(3,2)$ is transformed by the matrix $T=\begin{pmatrix}1 & 2\\ 0 & 1\end{pmatrix}.$ What are the coordinates of $S'$ ?
Answer: $(7,2)$ 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' = [1*3 + 2*2; 0*3 + 1*2] = [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!
A horizontal scaling by factor $2$ (double the $x$ -coordinate, keep $y$ the same) is represented by a matrix $H$ . What is the image of $S(3,-4)$ under $H$ ?
Answer: $(6,-4)$ 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 horizontal scaling by factor 2 doubles the x-coordinate while keeping y unchanged, so the matrix is H = [2 0; 0 1]. Applying this to point S(3, -4): H[3; -4] = [2 0; 0 1][3; -4] = [2·3 + 0·(-4); 0·3 + 1·(-4)] = [6 + 0; 0 + (-4)] = [6; -4], giving the image point (6, -4). Choice A correctly shows this result where x is doubled (3 → 6) and y stays the same (-4 → -4). Choice B incorrectly doubles the y-coordinate too, choice C halves x instead of doubling, and choice D doubles both coordinates. Matrix multiplication for transformations: For non-uniform scaling, use different diagonal entries [a 0; 0 b] where a scales x-coordinates and b scales y-coordinates. Horizontal scaling by 2 uses [2 0; 0 1], stretching horizontally while preserving vertical distances!
Which transformation is represented by the matrix $A=\begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}$ ?
Answer: Reflection across the $x$ -axis 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 = $\begin{pmatrix} \cos(\theta) & -\sin(\theta) \\ \sin(\theta) & \cos(\theta) \end{pmatrix}$ (example: 90° rotation uses θ=90° giving $\begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}$ since cos(90°)=0 and sin(90°)=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). 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 $±1$ on diagonal with one negative; test with a point like (1,1) to see the effect, and you'll master this quickly!
Rotate $90^\circ$ counterclockwise, then scale by a factor of $2$ . Using $R=\begin{bmatrix}0&-1\\1&0\end{bmatrix},\quad S=\begin{bmatrix}2&0\\0&2\end{bmatrix},$ what is the combined transformation matrix $S\cdot R$ ?
Answer: $\begin{bmatrix}0&-2\\2&0\end{bmatrix}$ 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! To combine rotation then scaling, compute S · R = [2 0; 0 2] · [0 -1; 1 0] = [0 -2; 2 0], applying rotation first and then scaling. Choice A correctly performs the matrix multiplication in the proper order for the composition. A distractor like choice B might swap signs by multiplying in the wrong order or miscalculating 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). Fantastic effort on compositions!
Triangle $ABC$ has vertices $A(1,1)$ , $B(3,1)$ , and $C(2,3)$ on the coordinate plane. The triangle is transformed by $S = \begin{pmatrix} 2 & 0 \\ 0 & 2 \end{pmatrix}$ . Which set of vertices gives the image triangle $A'B'C'$ ?
Answer: $A'(2,2), B'(6,2), C'(4,6)$ 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 = $[\cos(\theta) -\sin(\theta); \sin(\theta)\ \cos(\theta)]$ (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). Applying $S=[2\ 0; 0\ 2]$ 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!
A shear transformation $S = \begin{pmatrix} 1 & k \\ 0 & 1 \end{pmatrix}$ is applied to a parallelogram with base 4 and height 3. After transformation, one vertex moves from $(2,1)$ to $(5,1)$ . What is the area of the transformed parallelogram?
Answer: $12$ units² because shear transformations preserve all area measurements 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.
A point $P(x,y)$ is first reflected across the $x$ -axis and then reflected across the $y$ -axis. Let $F_x=\begin{pmatrix}1 & 0\\ 0 & -1\end{pmatrix},\quad F_y=\begin{pmatrix}-1 & 0\\ 0 & 1\end{pmatrix}.$ What single matrix represents the combined transformation (apply $F_x$ first, then $F_y$ )?
Answer: $\begin{pmatrix}-1 & 0\\ 0 & -1\end{pmatrix}$ 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: $T \begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} a & b \\ c & d \end{pmatrix} \begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} a x + b y \\ c x + d y \end{pmatrix} = \begin{pmatrix} x' \\ y' \end{pmatrix}$ 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° rotation uses $\theta=90°$ giving $\begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}$ since $\cos(90°)=0$ and $\sin(90°)=1$ ), 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: apply Fx first then Fy, so matrix = Fy * Fx = $\begin{pmatrix} -1 & 0 \\ 0 & 1 \end{pmatrix}$ * $\begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}$ = $\begin{pmatrix} -1 & 0 \\ 0 & -1 \end{pmatrix}$ , equivalent to 180° rotation. Choice C correctly computes this combined matrix. Distractors like choice A might give just one reflection matrix. For compositions, always multiply second times first and test with a point to confirm the net effect— you're doing fantastically!
Which matrix represents a rotation of $180^\circ$ about the origin?
Answer: $\begin{pmatrix}-1&0\\0&-1\end{pmatrix}$ 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!
Let $R=\begin{pmatrix}0 & -1\\ 1 & 0\end{pmatrix}$ (rotate $90^\circ$ counterclockwise) and $F=\begin{pmatrix}1 & 0\\ 0 & -1\end{pmatrix}$ (reflect across the $x$ -axis). What is the matrix for "rotate first, then reflect," i.e., $F\,R$ ?
Answer: $\begin{pmatrix}0 & -1\\ -1 & 0\end{pmatrix}$ 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. To find "rotate first, then reflect," we compute F·R = [1 0; 0 -1]·[0 -1; 1 0] = [1·0 + 0·1, 1·(-1) + 0·0; 0·0 + (-1)·1, 0·(-1) + (-1)·0] = [0, -1; -1, 0], giving the matrix [0 -1; -1 0]. Choice B correctly shows [0 -1; -1 0] as the composition of rotating 90° counterclockwise then reflecting across the x-axis. This combined transformation takes (1, 0) to (0, 1) after rotation, then to (0, -1) after reflection; and takes (0, 1) to (-1, 0) after rotation, which stays at (-1, 0) after x-axis reflection. Matrix multiplication for transformations: For composition F·R, multiply [1 0; 0 -1]·[0 -1; 1 0]. First column: [1 0; 0 -1]·[0; 1] = [0; -1]. Second column: [1 0; 0 -1]·[-1; 0] = [-1; 0]. Result: [0 -1; -1 0]. The order matters—R·F would give a different result!
What plane transformation is represented by the matrix $A=\begin{pmatrix}1 & 0\\ 0 & -1\end{pmatrix}?$
Answer: Reflection across the $x$ -axis 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!
Which matrix represents a reflection across the $y$ -axis?
Answer: $\begin{pmatrix}-1 & 0\\ 0 & 1\end{pmatrix}$ 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!
Let $T=\begin{bmatrix}-1&0\\0&1\end{bmatrix}.$ Apply $T$ to the point $(-5,2)$ . What is the image point?
Answer: $(5,2)$ 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 $\begin{bmatrix} x \\ y \end{bmatrix}$ and multiply by transformation matrix $T = \begin{bmatrix} a & b \\ c & d \end{bmatrix}$ using matrix multiplication: $T \begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} a & b \\ c & d \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} ax + by \\ cx + dy \end{bmatrix} = \begin{bmatrix} x' \\ y' \end{bmatrix}$ where $(x', y')$ is the transformed point. Common transformation matrices include: ROTATION by angle $θ$ counterclockwise = $\begin{bmatrix} \cos(θ) & -\sin(θ) \\ \sin(θ) & \cos(θ) \end{bmatrix}$ (example: 90° rotation uses $θ=90°$ 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! Applying $T = \begin{bmatrix} -1 & 0 \\ 0 & 1 \end{bmatrix}$ to $(-5, 2)$ gives $[-1*(-5) + 0*2; 0*(-5) + 1*2] = \begin{bmatrix} 5 \\ 2 \end{bmatrix}$ , so the image is $(5, 2)$ . 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 $\begin{bmatrix} a & b \\ c & d \end{bmatrix}$ 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 = $a·x + b·y$ . Second row times vector gives y'-coordinate = $c·x + d·y$ . (3) Result is transformed point $(x', y') = (ax+by, cx+dy)$ . Example: $\begin{bmatrix} 0 & -1 \\ 1 & 0 \end{bmatrix}$ 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 $\begin{bmatrix} \cos(θ) & -\sin(θ) \\ \sin(θ) & \cos(θ) \end{bmatrix}$ —look for this pattern with cos and -sin in first row, sin and cos in second row. Common: $\begin{bmatrix} 0 & -1 \\ 1 & 0 \end{bmatrix}$ is 90° rotation, $\begin{bmatrix} -1 & 0 \\ 0 & -1 \end{bmatrix}$ is 180° rotation. REFLECTION matrices have form $\begin{bmatrix} ±1 & 0 \\ 0 & ±1 \end{bmatrix}$ 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). Great job on this reflection transformation!
A rotation of $180^\circ$ about the origin is applied to point $N(6,-1)$ . What is the image of $N$ ?
Answer: $(-6,1)$ 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!
Triangle $ABC$ has vertices $A(1,1)$ , $B(3,1)$ , and $C(2,4)$ . It is transformed by the matrix $K=\begin{pmatrix}2 & 0\\ 0 & 2\end{pmatrix}.$ What are the coordinates of $C'$ (the image of $C$ )?
Answer: $(4,8)$ 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!
Which transformation is represented by the matrix $T=\begin{pmatrix}1&0\\0&-1\end{pmatrix}$ ?
Answer: Reflection across the $x$ -axis 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!
A uniform scaling by factor $3$ about the origin is applied to point $Q(-2,5)$ . Using a $2\times 2$ matrix, what is the image of $Q$ ?
Answer: $(-6,15)$ 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 uniform scaling by factor 3, the transformation matrix is S = [3 0; 0 3], which multiplies both coordinates by 3. Applying this to point Q(-2, 5): S[−2; 5] = [3 0; 0 3][−2; 5] = [3·(−2) + 0·5; 0·(−2) + 3·5] = [−6 + 0; 0 + 15] = [−6; 15], giving the image point (−6, 15). Choice A correctly shows this result where both coordinates are multiplied by 3: x' = 3·(−2) = −6 and y' = 3·5 = 15. Choice B incorrectly divides the y-coordinate instead of multiplying, choice C shows incorrect calculations, and choice D has the wrong signs. Matrix multiplication for transformations: SCALING matrices have equal diagonal entries [k 0; 0 k]—both coordinates multiplied by same k. For scaling by factor 3, use [3 0; 0 3], which enlarges all distances from the origin by factor 3!
The matrix $A=\begin{pmatrix}-1&0\\0&1\end{pmatrix}$ is applied to a point $(x,y)$ using $A\begin{pmatrix}x\\y\end{pmatrix}$ . Which description matches this transformation?
Answer: Reflection across the $y$ -axis 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° rotation uses $\theta=90°$ giving $\begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}$ since $\cos(90°)=0$ and $\sin(90°)=1$ ), 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! The matrix A = $\begin{pmatrix} -1 & 0 \\ 0 & 1 \end{pmatrix}$ transforms $(x, y)$ to $(-x, y)$ , negating the x-coordinate while keeping y the same, which is a reflection across the y-axis. Choice B correctly describes this reflection, as the negative diagonal entry affects only the x-component. A distractor like choice A might swap the axes, but note that x-axis reflection would negate y instead with $\begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}$ —double-check which coordinate is negated. 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° 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° rotation, $\begin{pmatrix} -1 & 0 \\ 0 & -1 \end{pmatrix}$ is 180° 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). Great job identifying reflections—practice with test points like $(1,0)$ to confirm!
A scaling transformation multiplies both coordinates by $3$ . Which matrix represents this transformation?
Answer: $\begin{bmatrix}3&0\\0&3\end{bmatrix}$ 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!
Triangle $\triangle PQR$ has vertices $P(1,1)$ , $Q(3,1)$ , and $R(2,3)$ . A transformation is given by $T=\begin{pmatrix}2 & 0\\ 0 & 2\end{pmatrix}.$ What are the coordinates of the image triangle $\triangle P'Q'R'$ ?
Answer: $P'(2,2),\ Q'(6,2),\ R'(4,6)$ 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!

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