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Solutions and invertibility

If the matrix \(A\) in the matrix equation

\[\begin{equation*}
  A \vec{x} = \vec{b}
\end{equation*}
\]

is invertible, then a unique solution

\[\begin{equation*}
  \vec{x} = A^{-1} \vec{b}
\end{equation*}
\]

exists for each \(\vec{b} \in \mathbb{R}^n\).