📘 Question
Given:
\[
I =
\begin{bmatrix}
1 & 0 \\
0 & 1
\end{bmatrix}, \quad
J =
\begin{bmatrix}
0 & 1 \\
-1 & 0
\end{bmatrix}
\]
and
\[
B =
\begin{bmatrix}
\cos\theta & \sin\theta \\
-\sin\theta & \cos\theta
\end{bmatrix}
\]
Express \(B\) in terms of \(I\) and \(J\).
✏️ Step-by-Step Solution
Step 1: Write combination
Observe:
\[
\cos\theta \cdot I =
\begin{bmatrix}
\cos\theta & 0 \\
0 & \cos\theta
\end{bmatrix}
\]
\[
\sin\theta \cdot J =
\begin{bmatrix}
0 & \sin\theta \\
-\sin\theta & 0
\end{bmatrix}
\]
Step 2: Add them
\[
\cos\theta I + \sin\theta J =
\begin{bmatrix}
\cos\theta & \sin\theta \\
-\sin\theta & \cos\theta
\end{bmatrix}
\]
Step 3: Compare
This matches matrix \(B\).
✅ Final Answer
\[
\boxed{B = \cos\theta \, I + \sin\theta \, J}
\]
💡 Key Concept
Matrix \(J\) behaves like a rotation generator. So rotation matrix can be written as:
\[
B = \cos\theta I + \sin\theta J
\]