For Any \(N \times N\) Square Matrix \(A\) if there exists such a Matrix \(B\) such that \(AB=BA=I\), (where \(I\) is the \(N \times N\) Identitity Matrix),
then the Matrix \(B\) is known as Inverse of Matrix \(A\)(denoted as \(A^{-1}\)).
For Any \(N \times N\) Square Matrix \(A\), \(A^{-1}\) Exists Only If Determinant of \(A\) is Not 0 (i.e. \(|A|\neq0\)).
The Inverse of Inverse of a Matrix gives back the Original Matrix. That is, if \(B=A^{-1}\), then \(A=B^{-1}\).
The Inverse of a Matrix \(A\) (\(A^{-1}\)) can be Calculated by using the Adjoint and Determinant of the Matrix as given in the following formula
where \(X=\) Unknown Matrix that is Inverse of the Matrix \(A\) (\(A^{-1}\)) , \(I=\)Identity Matrix
After Elementary Row/Column Operations are performed on Matrix \(A\) on the Left Hand Side of the Matrix Equation, it gets converted to Identity Matrix \(I\) and Matrix on the Right Hand Side of the Matrix Equation (which was previously the Identity Matrix \(I\)) gives the Inverse of the Matrix \(A\) (\(A^{-1}\)).
The Transpose of Inverse of a Matrix is same as Inverse of Transpose of the Matrix (\((A^{-1})^T=(A^T)^{-1}\)). This Matrix is also known as the Dual Matrix of the Matrix.
The Reciprocal/Inverse of Determinant Value of a Matrix is same as Determinant Value of Inverse of a Matrix. That is \(\frac{1}{|A|}=|A^{-1}|\)
The Inverse of a Matrix containing only a Single Non-Zero Value per Vector can be found out by Transponsing the Matrix and Replacing the Non-Zero Values with their Reciprocals. For example
\( \begin{bmatrix} 0 & 2 & 0\\ 0 & 0 & 7 \\ 8 & 0 & 0\end{bmatrix}^{-1}=\begin{bmatrix} 0 & 0 & \frac{1}{8}\\ \frac{1}{2} & 0 & 0 \\ 0 & \frac{1}{7} & 0\end{bmatrix}\)
The Inverse of any Orthogonal Matrix \(Q\) is same as its Transpose (i.e. \(Q^{-1}=Q^T\)).
The Inverse of any Unitary Matrix \(U\) is same as its Conjugate Transpose (i.e. \(U^{-1}=U^\dagger\)).