Using 2x2 Matrices for Plane Transformations - Precalculus

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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