### 2.10 Regular matrix, inverse matrix

Definition 26 if $$\det(\tilde A)\neq0$$ the $$\tilde A$$ is called a regular matrix, otherwise irregular or singular

inverse matrix:

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: $$b^{-1}a = ab^{-1}=\frac{a}{b}$$)

© J. Carstensen (Math for MS)