Lex Crameri

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.

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

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

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

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

• Algebra
• Perfectio quadri