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.