TITLE: Proof of the Inverse of a Scalar times a Matrix
QUESTION [18 upvotes]: How would I prove that given a square matrix $A$ and non-zero scalar $c$ that $$(cA)^{-1}=c^{-1}A^{-1}$$

REPLY [6 votes]: we want to prove $cA$ has inverse matrix $c^{-1}A^{-1}$. suppose $cA$ has inverse matrix $B$, that is we want to show $B = c^{-1}A^{-1}$. Here is the proof.
Since $B$ is the inverse matrix, then $(cA)B = I$, $c(AB) = I$, $AB = \frac{1}{c}I$, finally we multiply both sides with $A^{-1}$ on the left, $A^{-1}AB = A^{-1}\frac{1}{c}I$, we get $IB = \frac{1}{c}A^{-1}I = \frac{1}{c}A^{-1}$