Proof of the Inverse of a Scalar times a Matrix

Question

How would I prove that given a square matrix $A$ and non-zero scalar $c$ that $$(cA)^{-1}=c^{-1}A^{-1}$$

Answer 1

Assuming $A$ is invertible, so that $A^{-1}$ actually exists, you can simply check this directly. Is it true that $(cA)(c^{-1}A^{-1})=(c^{-1}A^{-1})(cA)=I$? Why?

Answer 2

Multiplying a matrix $A$by a constant $c$ is the same as scaling every row of a matrix by $c$. You can then consider $c$ to be the $ n \times n $ matrix $c Id$, (so that every entry in the diagonal equals $c$ and $0$ everywhere else, and where $A$ is also $n \times n$) and then $cId$ is invertible for all $c \neq 0$, and then apply the result that the inverse of $AB$ is $B^{-1}A^{-1}$.