Matrix Inverse

Summary

• \(A\) is a square \(n×n\) matrix. We say that \(A\) is invertible or nonsingular if there exists an \(n×n\) matrix \(B\) such that

\[AB=BA=I\]

• If no such matrix exist, we say that \(A\) is noninvertible or singular .
• If \(A\) is invertible, then the inverse of \(A\) is denoted \(A^{-1}\)
• If

\[A=\begin{bmatrix}a & b\\c & d\end{bmatrix}\]

• then \(A\) is invertible if and only if \(ad-cb≠0\) . If \(A\) is invertible then

\[A^{-1}= \frac{1}{ad-cb} \begin{bmatrix}d & -b\\-c & a\end{bmatrix}\]

• Results. \(A,B\) are \(n×n\) invertible matrices and \(α≠0\) is a scalar.
• \(A\) can have at most one inverse
• \((A^{-1})^{-1}=A\)
• \((αA)^{-1}=α^{-1}A^{-1}\)
• \((A^T)^{-1}=(A^{-1})^T\)
• \(AB\) is invertible and \((AB)^{-1}=B^{-1}A^{-1}\)