Precalculus Flashcards: Using 2x2 Matrices For Plane Transformations

What this deck covers

This deck focuses on Using 2x2 Matrices For Plane Transformations, giving you a quick way to review the definitions, rules, and examples that matter most for Precalculus.

How to use these flashcards

Work through these flashcards in short sessions. Try to answer each prompt before flipping the card, then revisit any cards you miss until the explanation feels automatic.

Flashcard 1: For $A=\begin{pmatrix}0&-1\\1&0\end{pmatrix}$, what is the image of $(x,y)$?

Answer: $(-y,\,x)$. This matrix rotates by $90°$ counterclockwise.

Flashcard 2: What is the matrix for a dilation by factor $k$ about the origin?

Answer: $\begin{pmatrix}k&0\\0&k\end{pmatrix}$. Scales both coordinates by factor $k$.

Flashcard 3: What is $\det\!\left(\begin{pmatrix}\cos\theta&-\sin\theta\\sin\theta&\cos\theta\end{pmatrix}\right)$?

Answer: $1$. $\cos^2\theta + \sin^2\theta = 1$, so determinant equals $1$.

Flashcard 4: What does a negative determinant ($\det(A)<0$) tell you about orientation?

Answer: Orientation is reversed (a reflection occurs). Negative determinant flips the plane's orientation.

Flashcard 5: Under $A=\begin{pmatrix}1&2\\0&1\end{pmatrix}$, what is the image of $(3,4)$?

Answer: $(11,4)$. $\begin{pmatrix}1&2\\0&1\end{pmatrix}\begin{pmatrix}3\\4\end{pmatrix}=\begin{pmatrix}3+8\\0+4\end{pmatrix}=\begin{pmatrix}11\\4\end{pmatrix}$.

Flashcard 6: What is the determinant formula for a $2\times 2$ matrix $\begin{pmatrix}a&b\\c&d\end{pmatrix}$?

Answer: $\det=ad-bc$. Multiply diagonal products and subtract.

Flashcard 7: What is the area of the parallelogram spanned by $u=(2,1)$ and $v=(5,3)$?

Answer: $|2\cdot 3-5\cdot 1|=1$. Area equals $|\det|$ of matrix with $u$ and $v$ as columns.

Flashcard 8: What is the image of $(2,-1)$ under $A=\begin{pmatrix}1&3\\0&-2\end{pmatrix}$?

Answer: $(-1,\,2)$. $A\begin{pmatrix}2\\-1\end{pmatrix} = \begin{pmatrix}1(2)+3(-1)\\0(2)+(-2)(-1)\end{pmatrix} = \begin{pmatrix}-1\\2\end{pmatrix}$.

Flashcard 9: What is the matrix for reflection across the $y$-axis?

Answer: $\begin{pmatrix}-1&0\\0&1\end{pmatrix}$. Negates $x$-coordinate while keeping $y$ unchanged.

Flashcard 10: What is the geometric meaning of $|\det(A)|$ for the unit square under $A$?

Answer: Area of the image parallelogram of the unit square. Unit square transforms to parallelogram with this area.

Flashcard 11: What is the image of $(x,y)$ under $A=\begin{pmatrix}a&b\\c&d\end{pmatrix}$?

Answer: $(x',y')=(ax+by,\,cx+dy)$. Matrix multiplication: first row gives $x'$, second row gives $y'$.

Flashcard 12: A region has area $6$. After $A$ with $\det(A)=-3$, what is the new area?

Answer: $18$. $|\det(A)| = |-3| = 3$, so area scales by $3$: $6 \times 3 = 18$.

Flashcard 13: For $A=\begin{pmatrix}3&0\\0&-2\end{pmatrix}$, what are $\det(A)$ and $|\det(A)|$?

Answer: $\det(A)=-6$ and $|\det(A)|=6$. For diagonal matrices, $\det =$ product of diagonal entries.

Flashcard 14: What does $|\det(A)|$ represent for a $2\times 2$ matrix acting on the plane?

Answer: Area scale factor: $\text{new area}=|\det(A)|\cdot\text{old area}$. The absolute value of the determinant scales areas by that factor.

Flashcard 15: What is the matrix for reflection across the line $y=x$?

Answer: $\begin{pmatrix}0&1\\1&0\end{pmatrix}$. Swaps $x$ and $y$ coordinates.

Flashcard 16: What is the area scale factor for $A=\begin{pmatrix}-2&0\\0&5\end{pmatrix}$?

Answer: $10$. $|\det| = |(-2)(5) - 0(0)| = |-10| = 10$.

Flashcard 17: What are the images of the basis vectors $(1,0)$ and $(0,1)$ under $A$?

Answer: $(1,0)\mapsto(a,c)$ and $(0,1)\mapsto(b,d)$. Columns of $A$ are images of standard basis vectors.

Flashcard 18: What is the area scale factor of a linear transformation with matrix $A$?

Answer: $|\det(A)|$. Absolute value of determinant measures how areas change.

Flashcard 19: What is the matrix for a reflection across the $y$-axis?

Answer: $\begin{pmatrix}-1&0\\0&1\end{pmatrix}$. Negates $x$, keeps $y$ same.

Flashcard 20: Find $\det\!\left(\begin{pmatrix}2&-1\\3&4\end{pmatrix}\right)$.

Answer: $11$. $2 \cdot 4 - (-1) \cdot 3 = 8 + 3 = 11$.

Flashcard 21: What is the matrix for a $90^\circ$ counterclockwise rotation about the origin?

Answer: $\begin{pmatrix}0&-1\\1&0\end{pmatrix}$. Maps $(1,0)$ to $(0,1)$ and $(0,1)$ to $(-1,0)$.

Flashcard 22: What is the matrix for reflection across the $x$-axis?

Answer: $\begin{pmatrix}1&0\\0&-1\end{pmatrix}$. Negates $y$-coordinate while keeping $x$ unchanged.

Flashcard 23: What is the matrix for a reflection across the $x$-axis?

Answer: $\begin{pmatrix}1&0\\0&-1\end{pmatrix}$. Keeps $x$ same, negates $y$.

Flashcard 24: What does $\det(A)<0$ indicate about a plane transformation?

Answer: Orientation is reversed (a reflection occurs). Negative determinant flips orientation of shapes.

Flashcard 25: What does $\det(A)=0$ tell you about the transformation in $\mathbb{R}^2$?

Answer: It collapses area to $0$ (not invertible). Zero determinant means the transformation is singular.

Flashcard 26: What is the determinant of $A=\begin{pmatrix}3&1\\2&4\end{pmatrix}$?

Answer: $10$. $\det = 3(4) - 1(2) = 12 - 2 = 10$.

Flashcard 27: What is the matrix for a $180^\circ$ rotation about the origin?

Answer: $\begin{pmatrix}-1&0\\0&-1\end{pmatrix}$. Negates both coordinates.

Flashcard 28: What does $\det(A)=0$ indicate about a plane transformation?

Answer: Area collapses to $0$ (not one-to-one; not invertible). Zero determinant means transformation squashes plane to a line.

Flashcard 29: What does a $2\times 2$ matrix $A$ do to a vector $(x,y)$ in the plane?

Answer: $A\begin{pmatrix}x\\y\end{pmatrix}$ gives the transformed vector. Matrix multiplication transforms the position vector.

Flashcard 30: What is the matrix for a horizontal shear with factor $k$ (so $(x,y)\mapsto(x+ky,y)$)?

Answer: $\begin{pmatrix}1&k\\0&1\end{pmatrix}$. Adds $k$ times $y$ to $x$, leaving $y$ unchanged.

Flashcard 31: What is the matrix for a horizontal shear with factor $k$ (maps $(x,y)$ to $(x+ky,y)$)?

Answer: $\begin{pmatrix}1&k\\0&1\end{pmatrix}$. Adds $k$ times $y$ to $x$-coordinate.

Flashcard 32: What is the matrix for a vertical shear with factor $k$ (maps $(x,y)$ to $(x,y+kx)$)?

Answer: $\begin{pmatrix}1&0\\k&1\end{pmatrix}$. Adds $k$ times $x$ to $y$-coordinate.

Flashcard 33: A region has area $8$. After applying $A$ with $\det(A)=\frac{1}{2}$, what is the new area?

Answer: $4$. New area = original area × $|\det(A)| = 8 × \frac{1}{2} = 4$.

Flashcard 34: If $\det(A)=-3$, by what factor does $A$ scale areas in the plane?

Answer: $3$. Area scale factor is $|\det(A)| = |-3| = 3$.

Flashcard 35: What is the matrix for a vertical shear with factor $k$ (so $(x,y)\mapsto(x,y+kx)$)?

Answer: $\begin{pmatrix}1&0\\k&1\end{pmatrix}$. Adds $k$ times $x$ to $y$, leaving $x$ unchanged.

Flashcard 36: What is the matrix for a counterclockwise rotation by angle $\theta$?

Answer: $\begin{pmatrix}\cos\theta&-\sin\theta\\sin\theta&\cos\theta\end{pmatrix}$. Standard rotation matrix rotates points counterclockwise by $\theta$.

Flashcard 37: What is the matrix for scaling by $s_x$ in $x$ and $s_y$ in $y$?

Answer: $\begin{pmatrix}s_x&0\\0&s_y\end{pmatrix}$. Diagonal matrix scales each axis independently.