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Lex Crameri est theorema algebrae linearis , quod systema aequationum 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.
Aequationum systemata in multiplicatione matricum sic representatur:
A
x
=
c
{\displaystyle Ax=c\,}
ubi matrix quadratus
A
{\displaystyle A}
x
{\displaystyle x}
(
x
i
)
{\displaystyle (x_{i})}
Theorema dicit:
x
i
=
det
(
A
i
)
det
(
A
)
{\displaystyle x_{i}={\det(A_{i}) \over \det(A)}}
ubi
A
i
{\displaystyle A_{i}}
i a columna
A
{\displaystyle A}
c
{\displaystyle c}
Lex Crameri in solvendo matricem 2×2 adhibetur, hac formula applicata:
Datum:
a
x
+
b
y
=
e
{\displaystyle ax+by=e\,}
c
x
+
d
y
=
f
{\displaystyle cx+dy=f\,}
quae in forma matricis:
(
a
b
c
d
)
(
x
y
)
=
(
e
f
)
{\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:
x
=
|
e
b
f
d
|
|
a
b
c
d
|
=
e
d
−
b
f
a
d
−
b
c
{\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
y
=
|
a
e
c
f
|
|
a
b
c
d
|
=
a
f
−
e
c
a
d
−
b
c
{\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
a
x
+
b
y
+
c
z
=
j
{\displaystyle ax+by+cz=j\,}
d
x
+
e
y
+
f
z
=
k
{\displaystyle dx+ey+fz=k\,}
g
x
+
h
y
+
i
z
=
l
{\displaystyle gx+hy+iz=l\,}
quae in forma matricis:
(
a
b
c
d
e
f
g
h
i
)
(
x
y
z
)
=
(
j
k
l
)
{\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:
x
=
|
j
b
c
k
e
f
l
h
i
|
|
a
b
c
d
e
f
g
h
i
|
{\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}}}}
y
=
|
a
j
c
d
k
f
g
l
i
|
|
a
b
c
d
e
f
g
h
i
|
{\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}}}}
z
=
|
a
b
j
d
e
k
g
h
l
|
|
a
b
c
d
e
f
g
h
i
|
{\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
