AP Precalculus Quiz: Matrices As Functions

What this quiz covers

This quiz focuses on Matrices As Functions, giving you a quick way to practice the rules, question types, and explanations that matter most for AP Precalculus.

How to use this quiz

Try each quiz question before looking at the correct answer. Use the explanations to review missed ideas, then come back to similar questions until the pattern feels familiar.

Question 1

1. Multiply by $\mathbf{A}^{-1}=\begin{pmatrix}\tfrac12&0\\0&2\end{pmatrix}$. (correct answer)
2. Add $\mathbf{I}=\begin{pmatrix}1&0\\0&1\end{pmatrix}$ to $\mathbf{A}$.
3. Multiply $\mathbf{A}$ by scalar $2$.
4. Transpose $\mathbf{A}$ to get $\mathbf{A}^T$.

Explanation: This question tests AP Precalculus understanding of matrices as functions, specifically focusing on inverse matrices and the identity matrix. The given matrix A = [[2,0],[0,1/2]] is diagonal, and its inverse is found by taking the reciprocal of each diagonal entry, giving A^(-1) = [[1/2,0],[0,2]]. When we multiply AA^(-1), we get the identity matrix I = [[1,0],[0,1]]. Choice A is correct because multiplying A by its inverse A^(-1) = [[1/2,0],[0,2]] yields the identity matrix. Choice B is incorrect as adding the identity matrix to A would give [[3,0],[0,3/2]], not the identity matrix, showing confusion between matrix operations. To help students: Practice finding inverses of diagonal matrices and verify results by multiplication. Watch for: Confusion between different matrix operations and misunderstanding what produces the identity matrix.

Question 2

1. $\mathbf{A}\mathbf{v}=\begin{pmatrix}6\\-4\end{pmatrix}$, a horizontal stretch by factor $3$. (correct answer)
2. $\mathbf{A}\mathbf{v}=\begin{pmatrix}5\\-4\end{pmatrix}$, a translation right by $3$.
3. $\mathbf{A}\mathbf{v}=\begin{pmatrix}6\\-12\end{pmatrix}$, a uniform scale by factor $3$.
4. $\mathbf{A}\mathbf{v}=\begin{pmatrix}2\\-1\end{pmatrix}$, a vertical shrink by factor $4$.

Explanation: This question tests AP Precalculus understanding of matrices as functions, specifically focusing on diagonal matrices and their geometric effects. The matrix A = [[3,0],[0,1]] scales the x-coordinate by 3 while leaving the y-coordinate unchanged, representing a horizontal stretch. When we multiply Av where v = [[2],[-4]], we get: first component = (3)(2) + (0)(-4) = 6, second component = (0)(2) + (1)(-4) = -4, resulting in [[6],[-4]]. Choice A is correct because it accurately computes Av = [[6],[-4]] and correctly identifies this as a horizontal stretch by factor 3. Choice C is incorrect as it miscalculates the y-component as -12 instead of -4, suggesting the student applied the x-scaling factor to both components. To help students: Visualize diagonal matrices as independent scaling of each axis and practice component-wise multiplication. Watch for: Students applying one scaling factor to all components instead of recognizing independent axis scaling.

Question 3

1. $\mathbf{A}\mathbf{v}=\begin{pmatrix}1\\5\end{pmatrix}$, found by adding vectors.
2. $\mathbf{A}\mathbf{v}=\begin{pmatrix}7\\3\end{pmatrix}$, found by row-by-column multiplication. (correct answer)
3. $\mathbf{A}\mathbf{v}=\begin{pmatrix}3\\7\end{pmatrix}$, found by swapping coordinates.
4. $\mathbf{A}\mathbf{v}=\begin{pmatrix}7\\1\end{pmatrix}$, found by multiplying entries directly.

Explanation: This question tests AP Precalculus understanding of matrices as functions, specifically focusing on matrix-vector multiplication using row-by-column method. For the matrix A = [[1,2],[0,1]] and vector v = [[1],[3]], we multiply: first row gives (1)(1) + (2)(3) = 1 + 6 = 7, second row gives (0)(1) + (1)(3) = 0 + 3 = 3, resulting in Av = [[7],[3]]. Choice B is correct because it accurately computes the matrix-vector product using proper row-by-column multiplication. Choice A is incorrect as it suggests adding vectors, showing fundamental misunderstanding of matrix multiplication - simply adding [[1,2]] and [[1,3]] component-wise would give [[2,5]], not [[1,5]]. To help students: Emphasize the row-by-column multiplication process systematically, working through each component calculation step by step. Watch for: Confusion between matrix addition and multiplication, and errors in applying the multiplication algorithm.

Question 4

1. $\det(\mathbf{A})=5$, so areas scale by a factor of $5$. (correct answer)
2. $\det(\mathbf{A})=-5$, so areas scale by a factor of $-5$.
3. $\det(\mathbf{A})=11$, so areas scale by a factor of $11$.
4. $\det(\mathbf{A})=6$, so areas scale by a factor of $6$.

Explanation: This question tests AP Precalculus understanding of matrices as functions, specifically focusing on determinant calculation and its geometric interpretation. The determinant of a 2×2 matrix represents how the matrix transformation scales areas, with the formula det(A) = ad - bc for matrix [[a,b],[c,d]]. For the given matrix A = [[2,1],[3,4]], we calculate det(A) = (2)(4) - (1)(3) = 8 - 3 = 5. Choice A is correct because it accurately states that det(A) = 5, meaning areas are scaled by a factor of 5 under this transformation. Choice C is incorrect as it miscalculates the determinant as 11, likely by adding all matrix entries instead of using the proper formula. To help students: Practice the determinant formula systematically and visualize how matrices transform unit squares to understand area scaling. Watch for: Students adding matrix entries or confusing the determinant formula with other matrix operations.

Question 5

1. A reflection across the $x$-axis. (correct answer)
2. A reflection across the $y$-axis.
3. A $90^\circ$ counterclockwise rotation.
4. A horizontal stretch by factor $-1$.

Explanation: This question tests AP Precalculus understanding of matrices as functions, specifically focusing on recognizing standard transformation matrices. The matrix A = [[1,0],[0,-1]] is a diagonal matrix where the x-component remains unchanged (multiplied by 1) while the y-component changes sign (multiplied by -1). This transformation takes any point (x,y) to (x,-y), which is precisely a reflection across the x-axis. Choice A is correct because it accurately identifies this transformation as a reflection across the x-axis. Choice B is incorrect as it would require the matrix [[-1,0],[0,1]], which negates x-coordinates instead of y-coordinates, a common confusion when students mix up axis reflections. To help students: Memorize standard transformation matrices and test them with simple points like (1,0) and (0,1) to verify the transformation. Watch for: Confusion between x-axis and y-axis reflections and misunderstanding which coordinate gets negated.

Question 6

1. $\mathbf{A}\mathbf{v}=\begin{pmatrix}3\\-2\end{pmatrix}$, a reflection across the $x$-axis.
2. $\mathbf{A}\mathbf{v}=\begin{pmatrix}-2\\3\end{pmatrix}$, a $90^\circ$ counterclockwise rotation. (correct answer)
3. $\mathbf{A}\mathbf{v}=\begin{pmatrix}2\\-3\end{pmatrix}$, a $90^\circ$ clockwise rotation.
4. $\mathbf{A}\mathbf{v}=\begin{pmatrix}-3\\-2\end{pmatrix}$, a $180^\circ$ rotation.

Explanation: This question tests AP Precalculus understanding of matrices as functions, specifically focusing on matrix multiplication and recognizing rotation transformations. The matrix A = [[0,-1],[1,0]] represents a 90° counterclockwise rotation transformation in the coordinate plane. When we multiply A by vector v = [[3],[2]], we perform row-by-column multiplication: first row gives (0)(3) + (-1)(2) = -2, second row gives (1)(3) + (0)(2) = 3, resulting in Av = [[-2],[3]]. Choice B is correct because it accurately computes the matrix-vector product and correctly identifies this as a 90° counterclockwise rotation. Choice C is incorrect as it reverses the result coordinates, suggesting a clockwise rotation instead, which is a common error when students confuse the rotation matrix forms. To help students: Practice recognizing standard transformation matrices, especially rotation matrices, and verify results by visualizing the geometric effect. Watch for: Confusion between clockwise and counterclockwise rotations and computational errors in matrix multiplication.

Question 7

1. It equals $-4$, so the transformation reverses orientation and quadruples area.
2. It equals $0$, so the transformation is not invertible and flattens area to zero. (correct answer)
3. It equals $4$, so the transformation is invertible and quadruples area.
4. It equals $1$, so the transformation preserves all lengths and angles.

Explanation: This question tests AP Precalculus understanding of matrices as functions, specifically focusing on determinant calculation and recognizing singular matrices. A zero determinant indicates the matrix is not invertible and collapses the plane to a line or point. For the matrix A = [[1,2],[2,4]], the determinant is calculated as (1)(4) - (2)(2) = 4 - 4 = 0. Choice B is correct because a determinant of 0 means the transformation is not invertible and flattens all areas to zero (the rows are linearly dependent). Choice C is incorrect as it claims the determinant is 4, missing that the rows are proportional. To help students: Emphasize that proportional rows always yield zero determinant and geometric collapse. Watch for: Students not recognizing when matrix rows are scalar multiples of each other.

Question 8

1. $\mathbf{A}\mathbf{v}=\begin{pmatrix}3\\4\end{pmatrix}$, adding vectors component-by-component.
2. $\mathbf{A}\mathbf{v}=\begin{pmatrix}7\\3\end{pmatrix}$, shearing by adding $2y$ into the $x$-value. (correct answer)
3. $\mathbf{A}\mathbf{v}=\begin{pmatrix}1\\5\end{pmatrix}$, shearing by adding $2x$ into the $y$-value.
4. $\mathbf{A}\mathbf{v}=\begin{pmatrix}2\\3\end{pmatrix}$, scaling the $x$-value by 2 only.

Explanation: This question tests AP Precalculus understanding of matrices as functions, specifically focusing on shear transformations through matrix multiplication. The matrix A = [[1,2],[0,1]] represents a horizontal shear that adds twice the y-coordinate to the x-coordinate. For v = [1,3], the multiplication yields: first row (1×1 + 2×3 = 1 + 6 = 7) and second row (0×1 + 1×3 = 3), giving [7,3]. Choice B is correct because it accurately describes the shearing effect where 2y is added to the x-value while y remains unchanged. Choice C is incorrect as it describes a vertical shear, which would have the form [[1,0],[2,1]]. To help students: Visualize shear transformations as slanting effects that preserve one coordinate while modifying the other. Watch for: Confusion between horizontal and vertical shears and their matrix representations.

Question 9

1. A reflection across the $y$-axis, sending $(x,y)$ to $(-x,y)$.
2. A $180^\circ$ rotation, sending $(x,y)$ to $(-x,-y)$.
3. A reflection across the $x$-axis, sending $(x,y)$ to $(x,-y)$. (correct answer)
4. A vertical stretch by factor $-1$, sending $(x,y)$ to $(x,y)$.

Explanation: This question tests AP Precalculus understanding of matrices as functions, specifically focusing on identifying reflection transformations from matrix form. The matrix A = [[1,0],[0,-1]] leaves x-coordinates unchanged while negating y-coordinates. This transformation sends any point (x,y) to (x,-y), which is the definition of reflection across the x-axis. Choice C is correct because the matrix precisely performs this x-axis reflection transformation. Choice A is incorrect as reflection across the y-axis would require the matrix [[-1,0],[0,1]], which negates x-coordinates instead. To help students: Have them apply the matrix to test points like (1,1) to verify the transformation type. Watch for: Confusion between x-axis and y-axis reflections and their corresponding matrices.

Question 10

1. $\det(\mathbf{A})=0$, so $\mathbf{A}$ is not invertible. (correct answer)
2. $\det(\mathbf{A})=8$, so $\mathbf{A}$ is not invertible.
3. $\det(\mathbf{A})=-8$, so $\mathbf{A}$ is invertible.
4. $\det(\mathbf{A})=2$, so $\mathbf{A}$ is invertible.

Explanation: This question tests AP Precalculus understanding of matrices as functions, specifically focusing on determinant calculation and its relationship to invertibility. For the matrix A = [[1,2],[2,4]], we calculate det(A) = (1)(4) - (2)(2) = 4 - 4 = 0. A matrix is invertible if and only if its determinant is non-zero, so a determinant of 0 means the matrix is singular (not invertible). Choice A is correct because it accurately states that det(A) = 0, therefore A is not invertible. Choice C is incorrect as it miscalculates the determinant as -8, possibly by reversing the subtraction in the formula, and incorrectly concludes the matrix is invertible. To help students: Emphasize that det(A) = 0 is the key indicator of non-invertibility, and practice recognizing when rows are scalar multiples. Watch for: Sign errors in determinant calculation and confusion about the relationship between determinant values and invertibility.

Question 11

1. $\mathbf{A}\mathbf{v}=\begin{pmatrix}3\\-2\end{pmatrix}$, a reflection across the $x$-axis.
2. $\mathbf{A}\mathbf{v}=\begin{pmatrix}-2\\3\end{pmatrix}$, a $90^\circ$ counterclockwise rotation. (correct answer)
3. $\mathbf{A}\mathbf{v}=\begin{pmatrix}2\\3\end{pmatrix}$, a $90^\circ$ clockwise rotation.
4. $\mathbf{A}\mathbf{v}=\begin{pmatrix}-3\\-2\end{pmatrix}$, a $180^\circ$ rotation.

Explanation: This question tests AP Precalculus understanding of matrices as functions, specifically focusing on matrix-vector multiplication and geometric transformations. The matrix A = [[0,-1],[1,0]] represents a 90° counterclockwise rotation transformation, which maps (x,y) to (-y,x). When we multiply Av where v = [[3],[2]], we get: first component = (0)(3) + (-1)(2) = -2, second component = (1)(3) + (0)(2) = 3, resulting in [[-2],[3]]. Choice B is correct because it accurately computes Av = [[-2],[3]] and correctly identifies this as a 90° counterclockwise rotation. Choice C is incorrect as it reverses the signs, suggesting a clockwise rotation instead, which is a common error when students misremember rotation matrices. To help students: Use the unit circle to verify rotation transformations and practice matrix multiplication step-by-step. Watch for: Sign errors in matrix multiplication and confusion between clockwise and counterclockwise rotations.

Question 12

1. Multiply by $\mathbf{A}^{-1}=\begin{pmatrix}\tfrac12&0\\0&\tfrac13\end{pmatrix}$ on either side. (correct answer)
2. Add $-\mathbf{A}$ to $\mathbf{A}$ to produce $\mathbf{I}$.
3. Transpose $\mathbf{A}$ so that $\mathbf{A}^T=\mathbf{I}$.
4. Multiply $\mathbf{A}$ by $\tfrac{1}{\det(\mathbf{A})}$ to get $\mathbf{I}$.

Explanation: This question tests AP Precalculus understanding of matrices as functions, specifically focusing on finding inverse matrices for diagonal matrices. The inverse of a diagonal matrix is found by taking the reciprocal of each diagonal entry. For the matrix A = [[2,0],[0,3]], the inverse is A^(-1) = [[1/2,0],[0,1/3]], and multiplying A by A^(-1) yields the identity matrix. Choice A is correct because multiplying by this inverse matrix (on either side) produces the identity matrix I. Choice D is incorrect as multiplying by 1/det(A) = 1/6 would give [[1/3,0],[0,1/2]], not the identity matrix. To help students: Emphasize that for diagonal matrices, the inverse has reciprocals on the diagonal. Watch for: Students confusing scalar multiplication with matrix inversion.

Question 13

1. It equals $-1$, so the transformation reverses orientation.
2. It equals $0$, so the transformation collapses the plane.
3. It equals $1$, so area is preserved and orientation stays the same. (correct answer)
4. It equals $2$, so all areas are doubled by the transformation.

Explanation: This question tests AP Precalculus understanding of matrices as functions, specifically focusing on determinant calculation and its geometric interpretation. The determinant of a 2×2 matrix represents how the transformation scales areas and whether it preserves or reverses orientation. For the matrix A = [[0,-1],[1,0]], the determinant is calculated as (0)(0) - (-1)(1) = 0 + 1 = 1. Choice C is correct because a determinant of 1 means the transformation preserves area (neither expanding nor contracting) and maintains the same orientation (positive determinant). Choice A is incorrect as it claims the determinant is -1, which would indicate orientation reversal. To help students: Practice calculating 2×2 determinants using the formula ad - bc and connect determinant values to geometric transformations. Watch for: Sign errors in determinant calculations and confusion between positive/negative determinant meanings.

Question 14

1. A reflection across the $x$-axis, sending $(x,y)$ to $(x,-y)$.
2. A reflection across the line $y=-x$, sending $(x,y)$ to $(-y,-x)$. (correct answer)
3. A $90^\circ$ counterclockwise rotation, sending $(x,y)$ to $(-y,x)$.
4. A horizontal stretch by factor $-1$, sending $(x,y)$ to $(-x,y)$.

Explanation: This question tests AP Precalculus understanding of matrices as functions, specifically focusing on identifying geometric transformations from matrix form. Matrices can represent various transformations like rotations, reflections, and stretches. The matrix A = [[0,-1],[-1,0]] transforms a point (x,y) to (-y,-x), which is characteristic of a reflection across the line y = -x. Choice B is correct because applying this matrix to any point (x,y) yields (-y,-x), which is exactly where the point lands after reflecting across y = -x. Choice C is incorrect as a 90° counterclockwise rotation would send (x,y) to (-y,x), not (-y,-x). To help students: Have them test the transformation on specific points like (1,0) and (0,1) to verify the geometric effect. Watch for: Confusion between different reflection lines and rotation angles.

Question 15

1. $\mathbf{A}\mathbf{v}=\begin{pmatrix}1\\-4\end{pmatrix}$, rotating the vector $90^\circ$ clockwise. (correct answer)
2. $\mathbf{A}\mathbf{v}=\begin{pmatrix}-1\\4\end{pmatrix}$, rotating the vector $90^\circ$ counterclockwise.
3. $\mathbf{A}\mathbf{v}=\begin{pmatrix}4\\1\end{pmatrix}$, leaving the vector unchanged.
4. $\mathbf{A}\mathbf{v}=\begin{pmatrix}5\\-3\end{pmatrix}$, adding coordinates as a vector sum.

Explanation: This question tests AP Precalculus understanding of matrices as functions, specifically focusing on matrix-vector multiplication representing rotations. The matrix A = [[0,1],[-1,0]] represents a 90° clockwise rotation transformation. For v = [4,1], the multiplication yields: first row (0×4 + 1×1 = 1) and second row (-1×4 + 0×1 = -4), giving [1,-4]. Choice A is correct because this result matches a 90° clockwise rotation of the vector [4,1]. Choice B is incorrect as it shows the result of a 90° counterclockwise rotation, which would use the matrix [[0,-1],[1,0]]. To help students: Connect rotation matrices to their geometric effects and practice verifying rotations by checking perpendicularity. Watch for: Confusion between clockwise and counterclockwise rotation matrices.

Question 16

1. It equals $-2$, so the transformation reverses orientation and doubles area.
2. It equals $2$, so the transformation preserves orientation and doubles area. (correct answer)
3. It equals $0$, so the transformation is not invertible and collapses the plane.
4. It equals $-8$, so the transformation reverses orientation and multiplies area by eight.

Explanation: This question tests AP Precalculus understanding of matrices as functions, specifically focusing on determinant calculation for non-diagonal matrices. The determinant provides information about area scaling and orientation preservation. For the matrix A = [[4,-2],[1,0]], the determinant is calculated as (4)(0) - (-2)(1) = 0 - (-2) = 2. Choice B is correct because a determinant of 2 means the transformation preserves orientation (positive) and doubles all areas. Choice D is incorrect as it miscalculates the determinant as -8, likely from an arithmetic error. To help students: Practice the determinant formula ad - bc carefully, paying attention to signs. Watch for: Sign errors when subtracting negative numbers in determinant calculations.

Question 17

1. $\det(\mathbf{A})=2$, so areas scale by a factor of $2$.
2. $\det(\mathbf{A})=6$, so areas scale by a factor of $6$. (correct answer)
3. $\det(\mathbf{A})=-6$, so areas scale by a factor of $-6$.
4. $\det(\mathbf{A})=5$, so areas scale by a factor of $5$.

Explanation: This question tests AP Precalculus understanding of matrices as functions, specifically focusing on determinant calculation and area scaling interpretation. For the matrix A = [[4,-1],[2,1]], we calculate det(A) = (4)(1) - (-1)(2) = 4 - (-2) = 4 + 2 = 6. The absolute value of the determinant represents the factor by which areas are scaled under the transformation. Choice B is correct because it accurately calculates det(A) = 6, meaning areas are scaled by a factor of 6. Choice C is incorrect as it miscalculates the determinant as -6, likely due to a sign error in the calculation, though the magnitude would still give the correct area scaling factor. To help students: Emphasize careful sign tracking in determinant calculations and remember that area scaling uses the absolute value of the determinant. Watch for: Sign errors when subtracting negative products in the determinant formula.

Question 18

1. A reflection across the line $y=x$. (correct answer)
2. A reflection across the $x$-axis.
3. A $90^\circ$ clockwise rotation about the origin.
4. A horizontal stretch by a factor of $1$.

Explanation: This question tests AP Precalculus understanding of matrices as functions, specifically focusing on identifying standard transformation matrices. The matrix A = [[0,1],[1,0]] swaps the x and y coordinates of any point, transforming (x,y) to (y,x), which geometrically represents a reflection across the line y = x. This can be verified by checking that points on the line y = x remain fixed, while points off the line are reflected across it. Choice A is correct because it accurately identifies this transformation as a reflection across the line y = x. Choice C is incorrect as it suggests a 90° clockwise rotation, which would map (x,y) to (y,-x), not (y,x), showing confusion between different transformation matrices. To help students: Test transformations on specific points and visualize the line of reflection by finding fixed points. Watch for: Confusion between reflections and rotations, especially when matrices have similar structures.

Question 19

1. $\mathbf{A}\mathbf{v}=\begin{pmatrix}5\\13\end{pmatrix}$, second output increases by $2$ times first. (correct answer)
2. $\mathbf{A}\mathbf{v}=\begin{pmatrix}10\\3\end{pmatrix}$, first output doubles from scalar multiplication.
3. $\mathbf{A}\mathbf{v}=\begin{pmatrix}8\\5\end{pmatrix}$, outputs are added component-wise.
4. $\mathbf{A}\mathbf{v}=\begin{pmatrix}5\\1\end{pmatrix}$, second output decreases by $2$ times first.

Explanation: This question tests AP Precalculus understanding of matrices as functions, specifically focusing on matrix multiplication in applied contexts. The matrix A = [[1,0],[2,1]] represents a transformation where the first output remains unchanged while the second output increases by 2 times the first input. When we multiply Av where v = [[5],[3]], we get: first component = (1)(5) + (0)(3) = 5, second component = (2)(5) + (1)(3) = 10 + 3 = 13, resulting in [[5],[13]]. Choice A is correct because it accurately computes Av = [[5],[13]] and correctly interprets that the second output (13) equals the original second input (3) plus 2 times the first input (2×5=10). Choice D is incorrect as it suggests the second output decreases, misunderstanding the positive coefficient in the matrix. To help students: Connect matrix operations to real-world models by interpreting each matrix entry's role in the transformation. Watch for: Misinterpretation of matrix coefficients and their effects on outputs in applied contexts.

Question 20

1. A reflection across the line $y=x$. (correct answer)
2. A reflection across the $x$-axis.
3. A $90^\circ$ clockwise rotation.
4. A dilation by factor $0$.

Explanation: This question tests AP Precalculus understanding of matrices as functions, specifically focusing on recognizing the matrix for reflection across y = x. The matrix A = [[0,1],[1,0]] swaps the x and y coordinates of any vector, transforming (x,y) to (y,x), which is precisely a reflection across the line y = x. This can be verified by noting that points on the line y = x remain fixed, while points are reflected across it. Choice A is correct because it accurately identifies this transformation as a reflection across the line y = x. Choice B is incorrect as a reflection across the x-axis would be represented by [[1,0],[0,-1]], which negates the y-coordinate rather than swapping coordinates. To help students: Test transformation matrices with specific points and visualize the geometric effect, particularly checking what happens to points like (1,0) and (0,1). Watch for: Confusion between different types of reflections and failure to recognize coordinate-swapping as reflection across y = x.