Lex Crameri

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Gabriel Cramerus
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Gabriel Cramerus

Lex Crameri est theorema algebrae linearis, quod systema aequatioum linearium per determinantes, a Gabriele Cramero (1704 - 1752) nominata.

Lex haud in computando est utile ergo rare aequationibus multis solvendis adhibetur. Tamen, algebrae theoriae importat, ut modum systematis solvendi explicate definit.

Index

[recensere] Formula simplex

Aequationum systemata in multiplicatione matricum sic representatur:

Ax = c\,

ubi matrix quadratus A invertier potest, et vector x est columnae vector mutabilum: (xi).

Theorema dicit:

x_i = { \det(A_i) \over \det(A)}

ubi Ai est matrix quae formatur ia columna A a columnae vectore c reposita.


[recensere] Exempla

Lex Crameri in solvendo matricem 2×2 adhibetur, hac formula applicata:

[recensere] 2X2

Datum:

ax + by = e\, et
cx + dy = f\,,

quae in forma matricis:

\begin{pmatrix} a & b \\ c & d \end{pmatrix}\begin{pmatrix} x \\ y \end{pmatrix}=\begin{pmatrix} e \\ f \end{pmatrix}

x et y possunt invenier lege Crameri:

x = \frac { \begin{vmatrix} e & b \\ f & d \end{vmatrix} } { \begin{vmatrix} a & b \\ c & d \end{vmatrix} } = { ed - bf \over ad - bc}
et
y = \frac { \begin{vmatrix} a & e \\ c & f \end{vmatrix} } { \begin{vmatrix} a & b \\ c & d \end{vmatrix} } = { af - ec \over ad - bc}


[recensere] 3x3

Lex matrici 3×3 est similis:

Datum

ax + by + cz = j\,,
dx + ey + fz = k\,, et
gx + hy + iz = l\,,

quae in forma matricis:

\begin{pmatrix} a & b & c \\ d & e & f \\ g & h & i \end{pmatrix}\begin{pmatrix} x \\ y \\ z \end{pmatrix}=\begin{pmatrix} j \\ k \\ l \end{pmatrix}

x, y, et z possunt invenier:

x = \frac { \begin{vmatrix} j & b & c \\ k & e & f \\ l & h & i \end{vmatrix} } { \begin{vmatrix} a & b & c \\ d & e & f \\ g & h & i \end{vmatrix} }, y = \frac { \begin{vmatrix} a & j & c \\ d & k & f \\ g & l & i \end{vmatrix} } { \begin{vmatrix} a & b & c \\ d & e & f \\ g & h & i \end{vmatrix} }, et z = \frac { \begin{vmatrix} a & b & j \\ d & e & k \\ g & h & l \end{vmatrix} } { \begin{vmatrix} a & b & c \\ d & e & f \\ g & h & i \end{vmatrix} }

[recensere] Vide etiam