Match each of the following transformations with its associated $$ 2 imes 2 $$ matrix. 1. Clockwise rotation by $$ \pi / 2 $$ radians 2. The projection onto the $$ x $$-axis given by $$ T(x, y)=(x, 0) $$ 3. Reflection about the $$ x $$-axis 4. Reflection about the line $$ y=x $$ 5. Counter-clockwise rotation by $$ \pi / 2 $$ radians 6. Reflection about the $$ y $$-axis A. $$ \left[\begin{array}{cc}1 & 0 \ 0 & 0\end{array} ight] $$ B. $$ \left[\begin{array}{cc}0 & -1 \ 1 & 0\end{array} ight] $$ C. $$ \left[\begin{array}{cc}-1 & 0 \ 0 & 1\end{array} ight] $$ D. $$ \left[\begin{array}{cc}0 & 1 \ 1 & 0\end{array} ight] $$ E. $$ \left[\begin{array}{cc}0 & 1 \ -1 & 0\end{array} ight] $$ F. $$ \left[\begin{array}{cc}1 & 0 \ 0 & -1\end{array} ight] $$ G. None of the above (Enter "A", "B", etc.; not "A.", "B.", etc.)

Question

Match each of the following transformations with its associated $$ 2 	imes 2 $$ matrix. 1. Clockwise rotation by $$ \pi / 2 $$ radians 2. The projection onto the $$ x $$-axis given by $$ T(x, y)=(x, 0) $$ 3. Reflection about the $$ x $$-axis 4. Reflection about the line $$ y=x $$ 5. Counter-clockwise rotation by $$ \pi / 2 $$ radians 6. Reflection about the $$ y $$-axis A. $$ \left[\begin{array}{cc}1 & 0 \ 0 & 0\end{array}ight] $$ B. $$ \left[\begin{array}{cc}0 & -1 \ 1 & 0\end{array}ight] $$ C. $$ \left[\begin{array}{cc}-1 & 0 \ 0 & 1\end{array}ight] $$ D. $$ \left[\begin{array}{cc}0 & 1 \ 1 & 0\end{array}ight] $$ E. $$ \left[\begin{array}{cc}0 & 1 \ -1 & 0\end{array}ight] $$ F. $$ \left[\begin{array}{cc}1 & 0 \ 0 & -1\end{array}ight] $$ G. None of the above (Enter "A", "B", etc.; not "A.", "B.", etc.)

UpStudy AI · Accepted Answer

Let's match each transformation with its associated $$ 2 \times 2 $$ matrix step by step:

1. **Clockwise rotation by $$ \pi / 2 $$ radians**: The matrix for a clockwise rotation by $$ \pi / 2 $$ radians is given by:
   $$
   \begin{bmatrix}
   0 & 1 \
   -1 & 0
   \end{bmatrix}
   $$
   This corresponds to **E**.

2. **The projection onto the $$ x $$-axis given by $$ T(x, y)=(x, 0) $$**: The matrix for this transformation is:
   $$
   \begin{bmatrix}
   1 & 0 \
   0 & 0
   \end{bmatrix}
   $$
   This corresponds to **A**.

3. **Reflection about the $$ x $$-axis**: The matrix for reflection about the $$ x $$-axis is:
   $$
   \begin{bmatrix}
   1 & 0 \
   0 & -1
   \end{bmatrix}
   $$
   This corresponds to **F**.

4. **Reflection about the line $$ y=x $$**: The matrix for reflection about the line $$ y=x $$ is:
   $$
   \begin{bmatrix}
   0 & 1 \
   1 & 0
   \end{bmatrix}
   $$
   This corresponds to **D**.

5. **Counter-clockwise rotation by $$ \pi / 2 $$ radians**: The matrix for a counter-clockwise rotation by $$ \pi / 2 $$ radians is:
   $$
   \begin{bmatrix}
   0 & -1 \
   1 & 0
   \end{bmatrix}
   $$
   This corresponds to **B**.

6. **Reflection about the $$ y $$-axis**: The matrix for reflection about the $$ y $$-axis is:
   $$
   \begin{bmatrix}
   -1 & 0 \
   0 & 1
   \end{bmatrix}
   $$
   This corresponds to **C**.

Now, summarizing the matches:
1. E
2. A
3. F
4. D
5. B
6. C

So the final answer is:
1. E
2. A
3. F
4. D
5. B
6. C
1. E 2. A 3. F 4. D 5. B 6. C

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