Let $A$ be a unital algebra over the complex numbers and $B$ be a subalgebra of $A$ with $A=B + \Bbb{C}1_A$. Suppose that $B$ has a unit $1_B \neq 1_A$ and that $\lambda \in \Bbb{C}\setminus \{0\}$. The book I'm reading claims:
$$b + \lambda 1_A \mathrm{\ invertible \ in \ A} \iff b + \lambda 1_B \mathrm{\ invertible \ in \ B} $$
I have trouble showing the implication $\implies $ . Suppose $b+ \lambda 1_A$ has inverse $c$ in $A$. What is the inverse of $b+ \lambda 1_B $ in $B$, or if an explicit inverse can't be written down how can one show it exists?
Suppose that $b+u1_A$ is invertible, $(b+u1_A)(b'+u'1_A)=1_A=bb'+u'b+ub'+uu'1_A$ since $B$ is a subalgebra $bb'+u'b+ub'=0, uu'=1$ and $(b+u1_B)(b'+u'1_B)=1_B$.