Symmetric group, matrix multiplication.

Question

One can associate an $n \times n$ matrix $M_\sigma$ with a permutation $\sigma \in S_n$ by letting the entry at $(i, \sigma(i))$ be $1$ and letting all other entries be $0$. For example, the matrix corresponding to the permutation$$\sigma = \begin{pmatrix} 1&2&3\\3&1&2\end{pmatrix} \in S_3$$would be$$M_\sigma = \begin{pmatrix}0 & 0 & 1\\0 & 1 & 0\\1 & 0 & 0\end{pmatrix}.$$Does it follow that $$M_{\sigma \tau} = M_\sigma M_\tau$$for all $\sigma, \tau \in S_n$, where the product on the right is the ordinary product of matrices?

Answer 1

Yes, for $\sigma\in S_n$, $M_\sigma$ is called a permutation matrix. Essentially it permutes the standard basis (that is, the columns of the identity matrix). So the actions of $M_{\sigma\tau}$ and $M_\sigma M_\tau$ on the standard basis coincide. In view of linear algebra, they define the same linear transformation.