Let $$\mathbf{A}=\left[\begin{array}{ccccc}a_{11} & a_{12} & a_{13} & \cdots & a_{1 n} \\a_{21} & a_{22} &a_{23} & \cdots & a_{2 n} \\a_{31} & a_{32} & a_{33} & \cdots & a_{3 n} \\\vdots & \vdots & \vdots & &\vdots \\a_{m 1} & a_{m 2} & a_{m 3} & \cdots & a_{m n}\end{array}\right]$$ $$\mathbf{C}=\left[\begin{array}{ccccc}c_{11} & c_{12} & c_{13} & \cdots & c_{1 n} \\c_{21} & c_{22} & c_{23} & \cdots & c_{2 n} \\c_{31} & c_{32} & c_{33} & \cdots & c_{3 n} \\\vdots & \vdots & \vdots & &\vdots \\c_{m 1} & c_{m 2} & c_{m 3} & \cdots & c_{m n}\end{array}\right]$$ and let $k$ and $l$ be any scalars.
$$\text { Prove that } k(\mathbf{A}+\mathbf{B})=k \mathbf{A}+k \mathbf{B}$$