Prove there exists $A \in M_n{(\mathbb R)}$ s.t. $A^2 = \begin{pmatrix} M & 0 \\ 0 & M \\ \end{pmatrix}$

Question

Prove that for all $M \in M_n{(\mathbb R)}$, there exists $A \in M_{2n}{(\mathbb R)}$ s.t. $A^2$ = $$\begin{pmatrix} M & 0 \\ 0 & M \\ \end{pmatrix} $$

I have tried by induction on $n$ but I don't see how to pass from $n$ to $n+1$.

Answer 1

A slightly different hint: $\begin{bmatrix}0&1\\1&0\end{bmatrix}^2=I.$

Answer 2

I think it's easier to do a concrete construction rather than induction. Try to find $A$ in block matrix form, i.e. similar to how you've written $A^2$.

For a further hint:

Have zero on the block diagonal.