Properties of invertible matrices
An \(n\times n\) matrix \(A\) is invertible if and only if \(A\sim I\text{.}\)
If \(A\) is invertible, then the solution to the equation \(A\xvec = \bvec\) is given by \(\xvec = A^{1}\bvec\text{.}\)

We can find \(A^{1}\) by finding the reduced row echelon form of \(\left[\begin{array}{rr} A \amp I \end{array}\right]\text{;}\) namely,
\begin{equation*} \left[\begin{array}{rr} A \amp I \end{array}\right] \sim \left[\begin{array}{rr} I \amp A^{1} \end{array}\right] \text{.} \end{equation*} If \(A\) and \(B\) are two invertible \(n\times n\) matrices, then their product \(AB\) is also invertible and \((AB)^{1} = B^{1}A^{1}\text{.}\)