Using 2x2 Matrices for Plane Transformations - Geometry

1. If $\det(A)=4$, what is the area scale factor of $A^{-1}$?

   $\frac{1}{4}$. Area scale factor is $|\frac{1}{4}| = \frac{1}{4}$.

2. A region has area $12$. What is its image area under a matrix with $\det(A)=\frac{1}{2}$?

   $6$. Area scales by $|\frac{1}{2}| = \frac{1}{2}$, so $12 \times \frac{1}{2} = 6$.

3. What is the determinant of $A=\begin{pmatrix}1&0\\0&-1\end{pmatrix}$?

   $-1$. $1 \times (-1) - 0 \times 0 = -1$

4. What is the determinant of a rotation matrix $R_\theta$?

   $\det(R_\theta)=1$. Rotations preserve areas, so determinant equals 1.

5. What is $\det\!\left(\begin{pmatrix}1&0\\k&1\end{pmatrix}\right)$ for a $y$-shear?

   $1$. Shears preserve area, so determinant is always 1.

6. A shape has area $7$. What is its image area under $A$ with $\det(A)=-3$?

   $21$. Area scales by $|-3| = 3$, so $7 \times 3 = 21$.

7. What is the $2\times 2$ matrix for reflection across the line $y=x$?

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

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

   $(x,\,kx+y)$. Shear shifts $y$ by $k$ times the $x$-coordinate.

9. What is $\det\!\left(\begin{pmatrix}k&0\\0&1\end{pmatrix}\right)$ for a horizontal stretch by $k$?

   $k$. Horizontal stretch scales area by factor $k$.

10. What is $\det\!\left(\begin{pmatrix}3&5\\2&4\end{pmatrix}\right)$?

    $2$. $3 \times 4 - 5 \times 2 = 12 - 10 = 2$

11. What is the area of the triangle spanned by $(1,2)$ and $(3,1)$ from one vertex?

    $\frac{5}{2}$. Triangle area is half the parallelogram area: $\frac{5}{2}$.

12. What is $\det(A^{-1})$ in terms of $\det(A)$ for an invertible $2\times 2$ matrix?

    $\det(A^{-1})=\frac{1}{\det(A)}$. Inverse transformation has reciprocal determinant.

13. What is the image of $(x,y)$ under $A=\begin{pmatrix}1&k\\0&1\end{pmatrix}$?

    $(x+ky,\,y)$. Shear shifts $x$ by $k$ times the $y$-coordinate.

14. What is the image of $(-5,2)$ under reflection across the $y$-axis?

    $(5,2)$. Reflection across $y$-axis negates $x$-coordinate.

15. What is the determinant of a uniform dilation matrix $kI$ in $\mathbb{R}^2$?

    $\det(kI)=k^2$. Uniform scaling in 2D scales area by $k^2$.

16. What is $\det\!\left(\begin{pmatrix}1&0\\0&k\end{pmatrix}\right)$ for a vertical stretch by $k$?

    $k$. Vertical stretch scales area by factor $k$.

17. What is the area scale factor for $A=\begin{pmatrix}2&0\\0&3\end{pmatrix}$?

    $|\det(A)|=6$. Area scale factor is the absolute value of determinant.

18. What is the area scale factor of the composition $AB$ in terms of determinants?

    $|\det(AB)|=|\det(A)|\,|\det(B)|$. Area scale factors multiply for composed transformations.

19. What is the area scale factor for $A=\begin{pmatrix}-2&0\\0&3\end{pmatrix}$?

    $|\det(A)|=6$. $|-2 \times 3| = 6$

20. What is the determinant of $A=\begin{pmatrix}0&-1\\1&0\end{pmatrix}$?

    $1$. $0 \times 0 - (-1) \times 1 = 1$

21. What is the determinant of any reflection matrix in the plane?

    $\det=-1$. Reflections preserve area but reverse orientation.

22. What is the matrix that sends $\vec{e}_1\mapsto(2,1)$ and $\vec{e}_2\mapsto(0,3)$?

    $\begin{pmatrix}2&0\\1&3\end{pmatrix}$. Columns are the images of standard basis vectors.

23. What is the matrix for a horizontal stretch by factor $k$ (scale $x$ only)?

    $\begin{pmatrix}k&0\\0&1\end{pmatrix}$. Stretches $x$-coordinate by factor $k$, leaves $y$ unchanged.

24. What is the determinant of $A=\begin{pmatrix}2&0\\0&3\end{pmatrix}$?

    $6$. $2 \times 3 - 0 \times 0 = 6$

25. What is $\det\!\left(\begin{pmatrix}0&1\\1&0\end{pmatrix}\right)$ for reflection across $y=x$?

    $-1$. $0 \times 0 - 1 \times 1 = -1$

26. What is the area of the parallelogram spanned by $(1,2)$ and $(3,1)$?

    $5$. $|1 \times 1 - 2 \times 3| = |1 - 6| = 5$

27. What does $|\det(A)|$ represent for a $2\times 2$ linear transformation of the plane?

    Area scale factor (multiplier) for all regions. The absolute value of determinant scales all areas uniformly.

28. What is the matrix for a shear parallel to the $x$-axis: $(x,y)\mapsto(x+ky,y)$?

    $\begin{pmatrix}1&k\\0&1\end{pmatrix}$. Shears horizontally: $x$ gets shifted by $k$ times $y$.

29. What condition on $\det(A)$ guarantees that a $2\times 2$ transformation is invertible?

    $\det(A)\neq 0$. Nonzero determinant means the transformation is one-to-one.

30. A region has area $5$. What is its image area under $A=\begin{pmatrix}1&4\\0&1\end{pmatrix}$?

    $5$. Shear has determinant 1, so area is preserved.

31. What is the image of $(-5,2)$ under reflection across the $x$-axis?

    $(-5,-2)$. Reflection across $x$-axis negates $y$-coordinate.

32. What is the orientation effect of $A=\begin{pmatrix}-1&0\\0&1\end{pmatrix}$?

    Orientation is reversed. Negative determinant indicates orientation reversal.

33. What is the area of the triangle with side vectors $\vec{u}$ and $\vec{v}$ from one vertex?

    $\frac{1}{2}\left|\det\!\left(\begin{pmatrix}u_1&v_1\\u_2&v_2\end{pmatrix}\right)\right|$. Triangle has half the area of its spanning parallelogram.

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

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

35. What is $\det\!\left(\begin{pmatrix}1&k\\0&1\end{pmatrix}\right)$ for an $x$-shear?

    $1$. Shears preserve area, so determinant is always 1.

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

    $(3,4)$. Second column gives image of $(0,1)$.

37. What is the area scale factor for reflection across $y=x$?

    $1$. Reflection has $|\det| = 1$, preserving all areas.

38. What is the $2\times 2$ matrix for a $180^\circ$ rotation about the origin?

    $\begin{pmatrix}-1&0\\0&-1\end{pmatrix}$. Rotates point $(x,y)$ to $(-x,-y)$ around origin.

39. What is the matrix for a vertical stretch by factor $k$ (scale $y$ only)?

    $\begin{pmatrix}1&0\\0&k\end{pmatrix}$. Stretches $y$-coordinate by factor $k$, leaves $x$ unchanged.

40. If $\det(A)=-2$ and $\det(B)=-3$, what is $\det(AB)$?

    $6$. $(-2) \times (-3) = 6$

41. If $|\det(A)|=2$ and $|\det(B)|=3$, what is $|\det(AB)|$?

    $6$. $2 \times 3 = 6$

42. What is the determinant of the identity matrix $I=\begin{pmatrix}1&0\\0&1\end{pmatrix}$?

    $\det(I)=1$. Identity transformation preserves all areas, so determinant is 1.

43. What is the matrix for a shear parallel to the $y$-axis: $(x,y)\mapsto(x,y+kx)$?

    $\begin{pmatrix}1&0\\k&1\end{pmatrix}$. Shears vertically: $y$ gets shifted by $k$ times $x$.

44. What is the area of the image of a region with area $K$ under matrix $A$?

    $K' = |\det(A)|\,K$. Area scales by the absolute value of the determinant.

45. What is the determinant of $A=\begin{pmatrix}a&b\\c&d\end{pmatrix}$?

    $\det(A)=ad-bc$. Formula: first diagonal product minus second diagonal product.

46. What is the $2\times 2$ matrix for a $90^\circ$ counterclockwise rotation?

    $\begin{pmatrix}0&-1\\1&0\end{pmatrix}$. Rotates point $(x,y)$ to $(-y,x)$ counterclockwise.

47. What is the area scale factor for $A=\begin{pmatrix}0&-2\\1&0\end{pmatrix}$?

    $|\det(A)|=2$. $|0 \times 0 - (-2) \times 1| = 2$

48. What is the image of $(4,7)$ under reflection across $y=x$?

    $(7,4)$. Reflection across $y=x$ swaps coordinates.

49. If $A\vec{e}_1=(a,c)$ and $A\vec{e}_2=(b,d)$, what is $A$?

    $\begin{pmatrix}a&b\\c&d\end{pmatrix}$. Matrix columns are images of basis vectors $\vec{e}_1, \vec{e}_2$.

50. What is the $2\times 2$ matrix for reflection across the $y$-axis?

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

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

    $A\begin{pmatrix}x\\y\end{pmatrix}=\begin{pmatrix}ax+by\\cx+dy\end{pmatrix}$. Matrix multiplication applies the transformation to the vector.

52. What does the sign of $\det(A)$ indicate for a plane transformation?

    $\det(A)>0$ preserves orientation; $\det(A)<0$ reverses it. Positive preserves clockwise/counterclockwise; negative flips it.

53. What is $\det(AB)$ in terms of $\det(A)$ and $\det(B)$ for $2\times 2$ matrices?

    $\det(AB)=\det(A)\det(B)$. Determinant of product equals product of determinants.

54. What is the $2\times 2$ matrix for a counterclockwise rotation by angle $\theta$?

    $\begin{pmatrix}\cos\theta&-\sin\theta\\sin\theta&\cos\theta\end{pmatrix}$. Standard rotation matrix using trigonometric identities.

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

    $(2,3)$. 90° rotation maps $(x,y)$ to $(-y,x)$.

56. What does $\det(A)=0$ imply about the image of the plane under $A$?

    It collapses to a line or point (not invertible). Zero determinant means the transformation is not invertible.

57. What is $\det\!\left(\begin{pmatrix}2&1\\0&3\end{pmatrix}\right)$?

    $6$. $2 \times 3 - 1 \times 0 = 6$

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

    $(2,1)$. First column gives image of $(1,0)$.

59. What is the $2\times 2$ matrix for a $270^\circ$ counterclockwise rotation?

    $\begin{pmatrix}0&1\\-1&0\end{pmatrix}$. Rotates point $(x,y)$ to $(y,-x)$ counterclockwise.

60. What is the $2\times 2$ matrix for reflection across the $x$-axis?

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

61. What is the area of the parallelogram spanned by vectors $\vec{u}$ and $\vec{v}$ in $\mathbb{R}^2$?

    $\left|\det\!\left(\begin{pmatrix}u_1&v_1\\u_2&v_2\end{pmatrix}\right)\right|$. Parallelogram area equals absolute value of cross product.

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