Definition 26 if \(\det(\tilde A)\neq0\) the \(\tilde A\) is called a regular matrix, otherwise irregular or singular
Definition 27
\(\tilde A\;N\times N\) Matrix with \(\det(\tilde A)\neq0\) then \(\tilde{A}^{-1}\)
defined by \(\tilde{A}\tilde{A}^{-1}\)\(=\tilde{A}^{-1}\tilde{A}=\tilde I\).
\(\tilde{A}^{-1}\) is called the inverse matrix with respect to \(\tilde{A}\). \(\tilde{A}^{-1}\)
is given by the following formula:
| \[\tilde{A}^{-1}=\frac{1}{\det(\tilde{A})}\left(\begin{array}{ccc}\backslash&&/\\&{A_{jk}}&\\/&&\backslash\end{array}\right)^\top_{k,j=1,\ldots,N},\] |
where \(A_{jk}\) are the cofactors of \(a_{jk}\) in \(\tilde A\).
Note: We still do not define a matrix division since
(\(\tilde{A},\tilde{B},\tilde{X}_1,\tilde{X}_2\) \(N\times N\) matrices, \(\det(\tilde{B})\neq0\))
\(\tilde{B}\tilde{X}_1=\tilde{A}\;\;\Rightarrow\;\;\tilde{X}_1=\tilde{B}^{-1}\tilde{A}\)
\(\tilde{X}_2\tilde{B}=\tilde{A}\;\;\Rightarrow\;\;\tilde{X}_2=\tilde{A}\tilde{B}^{-1}\neq\tilde{B}^{-1}\tilde{A}\) in general
\(\rightarrow\) thus, division cannot be properly defined!! (Scalars: \(b^{-1}a = ab^{-1}=\frac{a}{b}\))
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© J. Carstensen (Math for MS)