8.12 Matrix Inverse

The matrix \(\mathbf{B}\) such that \(\mathbf{AB} = \mathbf{BA} = \mathbf{I}\) (where \(\mathbf{I}\) is the identity matrix) is called the inverse of \(\mathbf{A}\) and is denoted by \(\mathbf{A}^{-1}\).

• If \(\mathbf{A}\) is a nonsingular square matrix, then there is a unique matrix \(\mathbf{B}\) such that \[\mathbf{AB} = \mathbf{BA} = \mathbf{I} \] To show the existence of an inverse, it is also equivalent to show that the determinant of \(\mathbf{A}\) is not zero.

• For square matrices \(\mathbf{A}\) and \(\mathbf{B}\) with the same dimension, \[(\mathbf{A}^{-1})^T = (\mathbf{A}^T)^{-1}\] \[(\mathbf{AB})^{-1} = \mathbf{B}^{-1}\mathbf{A}^{-1}\]