Question:

Let $\displaystyle A = [aij]$ be the n x n matrix defined by $\displaystyle a_{ii} = k$ and $\displaystyle A_{ij} = 0 $ if $\displaystyle i \neq j$. Show that if B is any n x n matrix, then $\displaystyle AB = kB.$.

My work:

$\displaystyle A = [aij]$ is not zero only when $\displaystyle i = j$, as $\displaystyle AB = kB$ would become $\displaystyle 0(B) = kB => 0 = 0$. Therefore, $\displaystyle A = [a_{ii}]$ since $\displaystyle j = i$ and $\displaystyle a_{ii} = k$. Hence, $\displaystyle AB => a_{ii}B => kB$.

What step am I missing? Although it is a relatively simple proof, I am still quite new at proof-writing. Thanks in advance.