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Wzór na macierz odwrotną 4x4

Mając macierz \(A\) taką że:

\(A=\begin{bmatrix}
a_{11} & a_{12} & a_{13} & a_{14}\\
a_{21} & a_{22} & a_{23} & a_{24}\\
a_{31} & a_{32} & a_{33} & a_{34}\\
a_{41} & a_{42} & a_{43} & a_{44}
\end{bmatrix}\)

Macierz odwrotną można obliczyć w następujący sposób:

jeżeli:

\(det(A)=a_{11}a_{22}a_{33}a_{44}+a_{11}a_{23}a_{34}a_{42}+a_{11}a_{24}a_{32}a_{43}+\)

\(\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:+a_{12}a_{21}a_{34}a_{43}+a_{12}a_{23}a_{31}a_{44}+a_{12}a_{24}a_{33}a_{41}+\)

\(\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:+a_{13}a_{21}a_{32}a_{44}+a_{13}a_{22}a_{34}a_{41}+a_{13}a_{24}a_{31}a_{42}+\)

\(\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:+a_{14}a_{21}a_{33}a_{42}+a_{14}a_{22}a_{31}a_{43}+a_{14}a_{23}a_{32}a_{41}+\)

\(\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:-a_{11}a_{22}a_{34}a_{43}-a_{11}a_{23}a_{32}a_{44}-a_{11}a_{24}a_{33}a_{42}+\)

\(\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:-a_{12}a_{21}a_{33}a_{44}-a_{12}a_{23}a_{34}a_{41}-a_{12}a_{24}a_{31}a_{43}+\)

\(\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:-a_{13}a_{21}a_{34}a_{42}-a_{13}a_{22}a_{31}a_{44}-a_{13}a_{24}a_{32}a_{41}+\)

\(\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:-a_{14}a_{21}a_{32}a_{43}-a_{14}a_{22}a_{33}a_{41}-a_{14}a_{23}a_{31}a_{42} \neq 0\)

to macierz odwrotna wynosi:

\(A^{-1}=\dfrac{1}{det(A)}\begin{bmatrix}
b_{11} & b_{12} & b_{13} & b_{14}\\
b_{21} & b_{22} & b_{23} & b_{24}\\
b_{31} & b_{32} & b_{33} & b_{34}\\
b_{41} & b_{42} & b_{43} & b_{44}
\end{bmatrix}\)

gdzie:

\(b_{11}=
a_{22}a_{33}a_{44}
+a_{23}a_{34}a_{42}
+a_{24}a_{32}a_{43}
-a_{22}a_{34}a_{43}
-a_{23}a_{32}a_{44}
-a_{24}a_{33}a_{42}\)

\(b_{12}=
a_{12}a_{34}a_{43}
+a_{13}a_{32}a_{44}
+a_{14}a_{33}a_{42}
-a_{12}a_{33}a_{44}
-a_{13}a_{34}a_{42}
-a_{14}a_{32}a_{43}\)

\(b_{13}=
a_{12}a_{23}a_{44}
+a_{13}a_{24}a_{42}
+a_{14}a_{22}a_{43}
-a_{12}a_{24}a_{43}
-a_{13}a_{22}a_{44}
-a_{14}a_{23}a_{42}\)

\(b_{14}=
a_{12}a_{24}a_{33}
+a_{13}a_{22}a_{34}
+a_{14}a_{23}a_{32}
-a_{12}a_{23}a_{34}
-a_{13}a_{24}a_{32}
-a_{14}a_{22}a_{33}\)

\(b_{21}=
a_{21}a_{34}a_{43}
+a_{23}a_{31}a_{44}
+a_{24}a_{33}a_{41}
-a_{21}a_{33}a_{44}
-a_{23}a_{34}a_{41}
-a_{24}a_{31}a_{43}\)

\(b_{22}=
a_{11}a_{33}a_{44}
+a_{13}a_{34}a_{41}
+a_{14}a_{31}a_{43}
-a_{11}a_{34}a_{43}
-a_{13}a_{31}a_{44}
-a_{14}a_{33}a_{41}\)

\(b_{23}=
a_{11}a_{24}a_{43}
+a_{13}a_{21}a_{44}
+a_{14}a_{23}a_{41}
-a_{11}a_{23}a_{44}
-a_{13}a_{24}a_{41}
-a_{14}a_{21}a_{43}\)

\(b_{24}=
a_{11}a_{23}a_{34}
+a_{13}a_{24}a_{31}
+a_{14}a_{21}a_{33}
-a_{11}a_{24}a_{33}
-a_{13}a_{21}a_{34}
-a_{14}a_{23}a_{31}\)

\(b_{31}=
a_{21}a_{32}a_{44}
+a_{22}a_{34}a_{41}
+a_{24}a_{31}a_{42}
-a_{21}a_{34}a_{42}
-a_{22}a_{31}a_{44}
-a_{24}a_{32}a_{41}\)

\(b_{32}=
a_{11}a_{34}a_{42}
+a_{12}a_{31}a_{44}
+a_{14}a_{32}a_{41}
-a_{11}a_{32}a_{44}
-a_{12}a_{34}a_{41}
-a_{14}a_{31}a_{42}\)

\(b_{33}=
a_{11}a_{22}a_{44}
+a_{12}a_{24}a_{41}
+a_{14}a_{21}a_{42}
-a_{11}a_{24}a_{42}
-a_{12}a_{21}a_{44}
-a_{14}a_{22}a_{41}\)

\(b_{34}=
a_{11}a_{24}a_{32}
+a_{12}a_{21}a_{34}
+a_{14}a_{22}a_{31}
-a_{11}a_{22}a_{34}
-a_{12}a_{24}a_{31}
-a_{14}a_{21}a_{32}\)

\(b_{41}=
a_{21}a_{33}a_{42}
+a_{22}a_{31}a_{43}
+a_{23}a_{32}a_{41}
-a_{21}a_{32}a_{43}
-a_{22}a_{33}a_{41}
-a_{23}a_{31}a_{42}\)

\(b_{42}=
a_{11}a_{32}a_{43}
+a_{12}a_{33}a_{41}
+a_{13}a_{31}a_{42}
-a_{11}a_{33}a_{42}
-a_{12}a_{31}a_{43}
-a_{13}a_{32}a_{41}\)

\(b_{43}=
a_{11}a_{23}a_{42}
+a_{12}a_{21}a_{43}
+a_{13}a_{22}a_{41}
-a_{11}a_{22}a_{43}
-a_{12}a_{23}a_{41}
-a_{13}a_{21}a_{42}\)

\(b_{44}=
a_{11}a_{22}a_{33}
+a_{12}a_{23}a_{31}
+a_{13}a_{21}a_{32}
-a_{11}a_{23}a_{32}
-a_{12}a_{21}a_{33}
-a_{13}a_{22}a_{31}\)