I have been practicing proof by induction, and attempted the following problem (question in red, my attempt in green). However, I am unsure with this is rigorous, and would greatly appreciate if someone can verify my proof, especially the inductive step. Thanks.
Your proof is fine.
If you wished you could make more use of the identity matrix viz.
$$A^{4(k+1)}=A^{4k}A^{4}=(-4)^kI(-4)I=(-4)^{k+1}I.$$