Prove that a matrix is a scalar matrix

Question

Given $k\in\mathbb{N}$, an invertible matrix $M$, and the equation:

$(M^{-1}AM)^k=3I$

I need to prove $A$ is a scalar matrix, without using eigenvalues. I understand why it's true, but can't prove it.

Answer 1

As noted in the other answers it is easy to show that $A^k$ must be a scalar matrix. But note that this does not means that $A$ is also a scalar matrix. As a counterexample use: $$ A=\begin{bmatrix} 1&1\\2&-1 \end{bmatrix} $$ for $k=2$.

Answer 2

We have $$aI = \left(M^{-1}AM\right)^k = \underbrace{\left(M^{-1}AM\right) \cdot \left(M^{-1}AM\right) \cdots \left(M^{-1}AM\right)}_{k \text{ times}} = M^{-1} A^k M \implies M(aI)M^{-1} = A^k$$ Hence, we have $$A^k = aI$$ Now conclude what you want.

Answer 3

Note that $M^{-1}AMM^{-1}AM= M^{-1}A^2M$, i.e. the 'interior' $M$s cancel out. By induction $$ (M^{-1}AM)^k = M^{-1}A^kM$$ Now multiply your original equation from the left with $M$ and from the right with it's inverse.