https://file.aiquickdraw.com/mathbot/file/1730820833752429pjdks.jpg

If I =
1 0
0 1
     , B =
0 1
1 0
       and
cos sin
sin cos
C
 
 
        then C=

To determine \( C \) in terms of \( I \) and \( B \), let's examine the provided matrices and verify if one of the given options matches the correct expression for \( C \).

Given matrices:
1. \( I = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} \), which is the identity matrix.
2. \( B = \begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix} \), which resembles a rotation matrix that represents a 90-degree rotation.
3. \( C = \begin{pmatrix} \cos \theta & \sin \theta \\ -\sin \theta & \cos \theta \end{pmatrix} \), which is a general rotation matrix by angle \( \theta \).

### Step 1: Rewrite \( C \) using \( I \) and \( B \)

The matrix \( C \) can be rewritten by recognizing it as a linear combination of \( I \) and \( B \):
\[
C = \cos \theta \cdot I + \sin \theta \cdot B
\]

This is because:
- \( I \) contributes the \( \cos \theta \) terms on the diagonal.
- \( B \) contributes the \( \sin \theta \) and \( -\sin \theta \) terms in the off-diagonal.

### Conclusion
Thus, the correct option is:
\[
C = I \cos \theta + B \sin \theta
\]

### Answer
The answer is **Option 1**: \( I \cos \theta + B \sin \theta \).

Would you like a more detailed breakdown or further questions to explore? Here are some related questions:

1. What are the properties of rotation matrices and how do they relate to \( I \) and \( B \)?
2. How can we express any rotation matrix as a combination of basic transformations?
3. Why does \( B \) represent a 90-degree rotation matrix?
4. How can we derive the general formula for combining \( I \) and \( B \) to represent rotations?
5. What happens if \( \theta = 90^\circ \) or \( \theta = 180^\circ \)? How does \( C \) simplify?

**Tip:** In linear algebra, combining transformations through matrix addition and scalar multiplication is a powerful method for representing complex transformations.

If \( I = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} \), \( B = \begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix} \), and \( C = \begin{pmatrix} \cos \theta & \sin \theta \\ -\sin \theta & \cos \theta \end{pmatrix} \), then what is \( C \)?

Learn how to express the rotation matrix \( C = \begin{pmatrix} \cos \theta & \sin \theta \\ -\sin \theta & \cos \theta \end{pmatrix} \) as a combination of matrices \( I \) and \( B \). This solution explores concepts in linear algebra and matrix operations, suitable for undergraduate students.

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