The inverse of a triangular matrix using determinants

Consider the lower triangular matrix A:
\left(\begin{array}{cccccc}a_{\mathrm{11}}&0&0&0&0&0\\ a_{\mathrm{21}}&a_{\mathrm{22}}&0&0&0&0\\ a_{\mathrm{31}}&a_{\mathrm{32}}&a_{\mathrm{33}}&0&0&0\\ a_{\mathrm{41}}&a_{\mathrm{42}}&a_{\mathrm{43}}&a_{\mathrm{44}}&0&0\\ a_{\mathrm{51}}&a_{\mathrm{52}}&a_{\mathrm{53}}&a_{\mathrm{54}}&a_{\mathrm{55}}&0\\ a_{\mathrm{61}}&a_{\mathrm{62}}&a_{\mathrm{63}}&a_{\mathrm{64}}&a_{\mathrm{65}}&a_{\mathrm{66}}\end{array}\right)

Modify as follows into matrix B:

\left(\begin{array}{cccccc}a_{\mathrm{11}}&0&0&0&0&1/(a_{\mathrm{11}}*a_{\mathrm{22}}*a_{\mathrm{33}}*a_{\mathrm{44}}*a_{\mathrm{55}}*a_{\mathrm{66}})\\ a_{\mathrm{21}}&a_{\mathrm{22}}&0&0&0&0\\ a_{\mathrm{31}}&a_{\mathrm{32}}&a_{\mathrm{33}}&0&0&0\\ a_{\mathrm{41}}&a_{\mathrm{42}}&a_{\mathrm{43}}&a_{\mathrm{44}}&0&0\\ a_{\mathrm{51}}&a_{\mathrm{52}}&a_{\mathrm{53}}&a_{\mathrm{54}}&a_{\mathrm{55}}&0\\ a_{\mathrm{61}}&a_{\mathrm{62}}&a_{\mathrm{63}}&a_{\mathrm{64}}&a_{\mathrm{65}}&0\end{array}\right)

Then calculate the determinant of matrix B and you will get the value at the sixth row and first column of the inverse of matrix A.

To calculate the value of the matrix inverse at the sixth row and second column calculate the determinant of the following matrix:

\left(\begin{array}{cccccc}a_{\mathrm{11}}&0&0&0&0&0\\ a_{\mathrm{21}}&a_{\mathrm{22}}&0&0&0&1/(a_{\mathrm{11}}*a_{\mathrm{22}}*a_{\mathrm{33}}*a_{\mathrm{44}}*a_{\mathrm{55}}*a_{\mathrm{66}})\\ a_{\mathrm{31}}&a_{\mathrm{32}}&a_{\mathrm{33}}&0&0&0\\ a_{\mathrm{41}}&a_{\mathrm{42}}&a_{\mathrm{43}}&a_{\mathrm{44}}&0&0\\ a_{\mathrm{51}}&a_{\mathrm{52}}&a_{\mathrm{53}}&a_{\mathrm{54}}&a_{\mathrm{55}}&0\\ a_{\mathrm{61}}&a_{\mathrm{62}}&a_{\mathrm{63}}&a_{\mathrm{64}}&a_{\mathrm{65}}&0\end{array}\right)

To calculate the value of the matrix inverse at the sixth row and third column calculate the determinant of the following matrix:

\left(\begin{array}{cccccc}a_{\mathrm{11}}&0&0&0&0&0\\ a_{\mathrm{21}}&a_{\mathrm{22}}&0&0&0&0\\ a_{\mathrm{31}}&a_{\mathrm{32}}&a_{\mathrm{33}}&0&0&1/(a_{\mathrm{11}}*a_{\mathrm{22}}*a_{\mathrm{33}}*a_{\mathrm{44}}*a_{\mathrm{55}}*a_{\mathrm{66}})\\ a_{\mathrm{41}}&a_{\mathrm{42}}&a_{\mathrm{43}}&a_{\mathrm{44}}&0&0\\ a_{\mathrm{51}}&a_{\mathrm{52}}&a_{\mathrm{53}}&a_{\mathrm{54}}&a_{\mathrm{55}}&0\\ a_{\mathrm{61}}&a_{\mathrm{62}}&a_{\mathrm{63}}&a_{\mathrm{64}}&a_{\mathrm{65}}&0\end{array}\right)

and so on.

See also: Inverse of a triangular matrix by matrix multiplication
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