Why do $\Gamma_{ij}^k\delta_{k}^{m}=\Gamma_{ij}^m$

Question

$$\Gamma_{ij}^k\delta_{k}^{m}=\Gamma_{ij}^m$$

Isn't it a scalar matrix?

Answer 1

Note that $$\Gamma_{ij}^k\delta_k^m=\sum_{k=1}^n\Gamma_{ij}^k\delta_k^m.$$ Now, $$\delta_k^m=\begin{cases}1, & k=m;\\ 0, &k\ne m.\end{cases}$$ Thus

$$\Gamma_{ij}^k\delta_k^m=\sum_{k=1}^n\Gamma_{ij}^k\delta_k^m=\Gamma_{ij}^m.$$