Similar Matrices - Matrix Multiplication How can we prove that $M A^3 M^{-1}$ is equal to $MAM^{-1}MAM^{-1}MAM^{-1}$ where M and A are square matrices and M is invertible ?

Start with $MAM^{-1}MAM^{-1}MAM^{-1}$ and simplify using associativity, inverses, and identity: \begin{align} MAM^{-1}MAM^{-1}MAM^{-1}&=MA(M^{-1}M)A(M^{-1}M)AM^{-1} \\\ &= MA(I)A(I)AM^{-1}\\\ &=MA^3 M^{-1} \end{align}