This problem involves determining a rotation matrix based on two sequential rotations applied to a coordinate frame. Here's the step-by-step approach:

1. Initial Condition: The frames $\{A\}$ and $\{B\}$ are initially coincident, meaning their coordinate axes are aligned.

2. First Rotation: The frame $\{B\}$ is rotated about its $\hat{Z}_B$-axis by an angle $\theta$.

3. Second Rotation: After the first rotation, the new orientation of frame $\{B\}$ is rotated about its $\hat{X}_B$-axis by an angle $\phi$.

4. Objective: To determine the rotation matrix that transforms a vector $\mathbf{P}$ described in frame $B$, denoted as $B\mathbf{P}$, to the same vector described in frame $A$, denoted as $A\mathbf{P}$.

Steps to Solve:

1. Rotation about $\hat{Z}_B$: A rotation about the $\hat{Z}$-axis by an angle $\theta$ is represented by the following rotation matrix: \cos\theta & -\sin\theta & 0 \\ \sin\theta & \cos\theta & 0 \\ 0 & 0 & 1 \end{pmatrix}$$
2. Rotation about $\hat{X}_B$: After the rotation by $\theta$, the frame is rotated about the new $\hat{X}_B$-axis by an angle $\phi$. A rotation about the $\hat{X}$-axis by $\phi$ is represented by: 1 & 0 & 0 \\ 0 & \cos\phi & -\sin\phi \\ 0 & \sin\phi & \cos\phi \end{pmatrix}$$
3. Combined Rotation: The combined rotation matrix that transforms from $B$ to $A$ is the product of the two rotations: $R = R_X(\phi) R_Z(\theta)$ where $R_Z(\theta)$ is applied first, followed by $R_X(\phi)$.

Let's compute the combined matrix $R$.