# How to prove $B=I+iA$, where $A$ is symmetric matrix, is invertible? [closed]

Some Background: There are many simple results in matrix theory which mimic and/or stem from analogous facts which hold for real or complex numbers. For example, complex numbers of the form $1 + ri$, $r \in \Bbb R$, are always invertible in $\Bbb C$: $(1 + ri)^{-1} = (1 - ri)/(1 + r^2)$. This problem shows how the invertibility of $1 + ri$ generalizes to allow $r$ to be replaced by a symmetric real matrix $A$, thus verifying a useful property which carries over to the matrix case. Note Added by Robert Lewis, 30 August 2018, 9:58 AM PST.

I have prove it in $2 \times 2$ case. but to generalize it $n \times n$ case.

• Please include your proof for the $2\times 2$ case, so we can see where it might generalize. Aug 30 '18 at 15:45
• Is $i=\sqrt {-1}$ here? What if we take $A=i\times I$?
– lulu
Aug 30 '18 at 15:45