1. **Stating the problem:** We want to multiply the scalar $\sqrt{x}$ by the matrix $\begin{bmatrix}a & b \\ c & d\end{bmatrix}$. 2. **Formula and rules:** When multiplying a scalar by a matrix, multiply each element of the matrix by the scalar. 3. **Intermediate work:** Multiply each element by $\sqrt{x}$: $$\sqrt{x} \times \begin{bmatrix}a & b \\ c & d\end{bmatrix} = \begin{bmatrix}\sqrt{x}a & \sqrt{x}b \\ \sqrt{x}c & \sqrt{x}d\end{bmatrix}$$ 4. **Explanation:** Each element of the matrix is scaled by $\sqrt{x}$, so the resulting matrix has each entry multiplied by $\sqrt{x}$. **Final answer:** $$\begin{bmatrix}\sqrt{x}a & \sqrt{x}b \\ \sqrt{x}c & \sqrt{x}d\end{bmatrix}$$