Proof of Theorem SNCM

By Theorem NMUS we know already that $\linearsystem{A}{\vect{b}}$ has a unique solution for every choice of $\vect{b}$. We need to show that the expression stated is indeed a solution (the solution). That is easy, just “plug it in” to the vector equation representation of the system (Theorem SLEMM), \begin{align*} A\left(\inverse{A}\vect{b}\right) &=\left(A\inverse{A}\right)\vect{b}&&\knowl{./knowls/theorem.MMA.knowl}{\text{Theorem MMA}}\\ &=I_n\vect{b}&&\knowl{./knowls/definition.MI.knowl}{\text{Definition MI}}\\ &=\vect{b}&&\knowl{./knowls/theorem.MMIM.knowl}{\text{Theorem MMIM}} \end{align*}

Since $A\vect{x}=\vect{b}$ is true when we substitute $\inverse{A}\vect{b}$ for $\vect{x}$, $\inverse{A}\vect{b}$ is a (the!) solution to $\linearsystem{A}{\vect{b}}$.