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Inverse Matrices

The inverse of a matrix is another matrix which, when multiplied by the first matrix, yields the unit matrix ${\bf I}$ (a matrix with all zeroes except 1's down the diagonal).
{\bf A}^{-1} {\bf A} = {\bf I}.
\end{displaymath} (22)

In the general case, the inverse may be written

{\bf A}^{-1} = \frac{1}{\vert{\bf A}\vert} {\bf A}_{cof}^T,
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where ${\bf A}_{cof}^T$ is the transpose of the matrix of cofactors $A_{ij} = (-1)^{i+j} \vert{\bf M}_{ij}\vert$. For example:
\left( \begin{array}{cc} a & b \\ c & d \end{array} \right)^...
...\left( \begin{array}{cc} d & -b \\ -c & a \end{array} \right).
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Clearly, there are major problems in finding the inverse of a matrix if it has a determinant equal to zero, since the formula for ${\bf A}^{-1}$ involves dividing by the determinant. In such cases, we say that ${\bf A}$ is singular and has no inverse.

David Sherrill 2006-08-15