Lex Crameri

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

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

Formula simplex

Aequationum systemata in multiplicatione matricum sic representatur:

${\displaystyle Ax=c\,}$

ubi matrix quadratus ${\displaystyle A}$ inverti potest, et vector ${\displaystyle x}$ est columnae vector mutabilum: ${\displaystyle (x_{i})}$.

Theorema dicit:

${\displaystyle x_{i}={\det(A_{i}) \over \det(A)}}$

ubi ${\displaystyle A_{i}}$ est matrix quae formatur i^a columna ${\displaystyle A}$ a columnae vectore ${\displaystyle c}$ reposita.

Exempla

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

2X2

Datum:

${\displaystyle ax+by=e\,}$ et

${\displaystyle cx+dy=f\,}$,

quae in forma matricis:

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

x et y possunt inveniri lege Crameri:

${\displaystyle x={\frac {\begin{vmatrix}e&b\\f&d\end{vmatrix}}{\begin{vmatrix}a&b\\c&d\end{vmatrix}}}={ed-bf \over ad-bc}}$

et

${\displaystyle y={\frac {\begin{vmatrix}a&e\\c&f\end{vmatrix}}{\begin{vmatrix}a&b\\c&d\end{vmatrix}}}={af-ec \over ad-bc}}$

3x3

Lex matrici 3×3 est similis:

Datum

${\displaystyle ax+by+cz=j\,}$,

${\displaystyle dx+ey+fz=k\,}$, et

${\displaystyle gx+hy+iz=l\,}$,

quae in forma matricis:

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

x, y, et z possunt inveniri:

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

Nexus interni

• Algebra
• Perfectio quadri