Is it possible to find matrix solutions to the following : $$(\sum_1^m M_k x_k)^n=\sum_1^m x_k^n$$ where $M_k$ are the desired $d \times d$ matrices (no restriction on $d$) and $x_i$ are indeterminate variables; For n=2 the gamma matrices satisfing $M_i M_j + M_j M_i = 2\delta_{ij}$ work; so in a way this is a generalization of these to larger $n$.

Is it possible to find matrix solutions to the following : $$\left(\sum_1^m M_k x_k\right)^n=\left(\sum_1^m x_k^n\right)I_d$$ where $M_k$ are the desired $d \times d$ matrices (no restriction on $d$) and $x_i$ are indeterminate variables; For n=2 the gamma matrices satisfing $M_i M_j + M_j M_i = 2\delta_{ij}$ work; so in a way this is a generalization of these to larger $n$.