matrix inverse
Matrix inverse
The inverse matrix is reciprocal in the sense that \[A^{-1}A=I\], see identity matrix. A matrix that can be inversed is known as a non-singular matrix while a matrix that cannot be inversed is known as a singular matrix (which essentially means it's determinant is 0).
In general \[(AB)^{-1}=B^{-1}A^{-1}\]. Note that \[(AB)^{-1}\ne A^{-1}B^{-1}\].
\[2\times 2\] matrix
In general, the inverse for a \[2 \times 2\] matrix, say \[A\] is:
The determinant of \[A\], \[\det(A)\], \[\det(A)=ad-bc\].
\[3\times 3\] matrix
See example in Gaussian elimination.
\[\det (A^{-1})=\det (A)^{-1}\]
We know that \[\det(AB)=\det(A)\cdot\det(B)\] and \[AA^{-1}=I\], \[\det(AA^{-1})=\det(I)\implies \det(A)\cdot\det(A^{-1})=\det(I)\]. Since \[\det(I)=1\] regardless of its dimensions, \[\det(A)\cdot\det(A^{-1})=1\] or \[\det(A^{-1})=\det(A)^{-1}\].