Matrix/Elementary row operations/Elementary matrix from left/Fact/Proof/Exercise

Let ${}K$ be a field and ${}M$ a ${}n\times n$-matrix with entries in ${}K$. Prove that the multiplication by the elementary matrices from the left with M has the following effects.

1. ${}V_{ij}\circ M=$ exchange of the ${}i$-th and the ${}j$-th row of ${}M$.
2. ${}(S_{k}(s))\circ M=$ multiplication of the ${}k$-th row of ${}M$ by ${}s$.
3. ${}(A_{ij}(a))\circ M=$ addition of ${}a$-times the ${}j$-th row of ${}M$ to the ${}i$-th row (${}i\neq j$).

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