To prove these results, we will use the definitions of matrix addition and scalar multiplication. Let A and B be m x n matrices, and let g and k be scalars.

First, calculate $A+B$.

Now, multiply by the scalar $k$.

Using the distributive property of scalars over real numbers, $k(a_{ij} + b_{ij}) = ka_{ij} + kb_{ij}$.
So,

First, calculate $kA$.

Next, calculate $kB$.

Now, add $kA$ and $kB$.

Since $[ka_{ij} + kb_{ij}] = [ka_{ij} + kb_{ij}]$, the LHS equals the RHS.
Therefore, $k(A+B) = kA + kB$ is proven.
$\boxed{\text{Proven}}$

Using the distributive property of scalars over real numbers, $(g+k)a_{ij} = ga_{ij} + ka_{ij}$.
So,

First, calculate $gA$.

Next, calculate $kA$.

Now, add $gA$ and $kA$.

Since $[ga_{ij} + ka_{ij}] = [ga_{ij} + ka_{ij}]$, the LHS equals the RHS.
Therefore, $(g+k)A = gA + kA$ is proven.
$\boxed{\text{Proven}}$