Página — Matematica 9° Secundaria (pág. 47)

Hallamos el menor de a_{_3}

\begin{vmatrix}a_{1}&b_{1}&c_{1}\\ a_{2}&b_{2}&c_{2}\\ \hline\quad&\quad&\quad\\ \hline\quad&\quad&c_{3}\end{vmatrix}\rightarrow\begin{vmatrix}b_{1}&c_{1}\\ b_{2}&c_{2}\end{vmatrix}\quad\begin{array}{l}\\ \end{vmatrix}\quad\begin{array}{l}\\ \end{array}\quad\begin{array}{l}\\ \end{array}\quad\begin{array}{l}\\ \end{array}\quad\begin{array}{l}\\ \end{array}\quad\begin{array}{l}\\ \end{array}\quad\begin{array}{l}\\ \end{array}\quad\begin{array}{l}\\ \end{array}\quad\begin{array}{l}\\ \end{array}\quad\begin{array}{l}\\ \end{array}\quad\begin{array}{l}\\ \end{array}\quad\begin{array}{l}\\ \end{array}\quad\begin{array}{l}\\ \end{array}\quad\begin{array}{l}\\ \end{array}\quad\begin{array}{l}\\ \end{array}\quad\begin{array}{l}\\ \end{array}\quad\begin{array}{l}\\ \end{array}\quad\begin{array}{l}\\ \end{array}\quad\begin{array}{l}\\ \end{array}\quad\begin{array}{l}\\ \end{array}\quad\begin{array}{l}\\ \end{array}\quad\begin{array}{l}\\ \end{array}\quad\begin{array}{l}\\ \end{array}\quad\begin{array}{l}\\ \end{array}\quad\begin{array}{l}\\ \end{array}\quad\begin{array}{l}\\ \end{array}\quad\begin{array}{l}\\ \end{array}\quad\begin{array}\begin{array}{l}\\ \end{array}\quad\begin{array}{l}\\ \end{array}\quad\begin{array}\begin{array}{l}\\ \end{array}\quad\begin{array}\begin{array}{l}\\ \end{array}\quad\begin{array}\begin{array}{l}\\ \end{array}\quad\begin{array}\begin{array}\begin{array}{l}\\ \end{array}\quad\begin{array}\begin{array}\end{array}\quad\begin{array}\begin{array}\end{array}\quad\begin{array}\begin{array}\end{array}\quad\begin{array}\begin{array}\begin{array}\end{l}\quad\begin{array}\begin{array}\end{array}\quad\begin{array}\begin{array}\end{array}\quad\begin{array}\begin{array}\end{array}\quad\begin{array}\begin{array}\end{array}\quad\begin{array}\begin{array}\begin{array}\end{array}\quad\begin{array}\begin{array}\begin{array}\end{array}\end{array}\quad\begin{array}\begin{array}\begin{array}\begin{array}\begin{array}\end{array}\begin{array}\end{array}\begin{array}\end{array}\begin{array}\begin{array}\begin{array}\begin{array}\begin{array}\end{array}\begin{array}\begin{array}\begin{array}\begin{array}\end{array}\begin{array}\begin{array}\end{array}\begin{array}\begin{array}\begin{array}\begin{array}\end{array}\begin{array}\begin{array}\begin{array}\begin{array}\begin{array}\begin{array}\begin{array}\end{array}\begin{array}\begin{array}\end{array}\begin{array}\begin{array}\begin{array}\begin{array}\begin{array}\begin{array}\begin{array}\begin{array}\end{array}\begin{array}\begin{array}\begin{array}\begin{array}\begin{array}\begin{array}\end{array}\begin{array}\begin{array}\end{array}\begin{array}\begin{array}\begin{array}\begin{array}\begin{array}\begin{array}\begin{array}\begin{array}\begin{array}\begin{array}\end{array}\begin{array}\begin{array}\end{array}\begin{array}\begin{array}\begin{array}\begin{array}\begin{array}\begin{array}\begin{array}\begin{array}\begin{array}\begin{array}\begin{array}\begin{array}\end{array}\begin{array}\begin{array}\begin{array}\begin{array}\end{array}\begin{array}\begin{array}\begin{array}\begin{array}\begin{array}\begin{array}\begin{array}\begin{array}\begin{array}\begin{array}\begin{array}\begin{array}\begin{array}\begin{array}\begin{array}\begin{array}\begin{array}\begin{array}\end{array}\begin{array}\begin{array}\begin{array}\begin{array}\begin{array}\begin{array}\begin{array}\begin{array}\begin{array}\begin{array}\begin{array}\begin{array}\begin{array}\begin{array}\begin{array}\begin{array}\begin{array}\begin{array}\begin{array}\begin{array}\begin{array}\begin{array}\end{array}\begin{array}\begin{array}\begin{array}\begin{array}\begin{array}\begin{array}\begin{array}\begin{array}\begin{array}\end{array}\begin{array}\begin{array}\begin{array}\begin{array}\begin{array}\begin{array}\end{array}\begin{array}\begin{array}\begin{array}\begin{array}\begin{array}\begin{array}\begin{array}\begin{array}\begin{array}\begin{array}\begin{array}\begin{array}\begin{array}\begin{array}\begin{array}\begin{array}\begin{array}\begin{array}\begin{array}\begin{array}\begin{array}\begin{array}\begin{array}\begin{array}\end{array}\begin{array}\begin{array}\begin{array}\begin{array}\begin{array}\begin{array}\begin{array}\begin{array}\begin{array}\begin{array}\begin{array}\begin{array}\begin{array}\begin{array}\begin{array}\begin{array}\begin{array}\begin{array}\begin{array}\begin{array}\begin{array}\begin{array}\begin{array}\begin{c}\begin{c}\begin{array}\begin{c}\begin{array}\begin{c}\begin{c}\begin{c}\begin{c}\begin{c}\begin{c}\end{c}\end{c}\end{c}\begin{c\end{c_{c_{c_{c_{c_{c_{c_{c_{c_{1\\ a_{1\\ a_{1\\ a_{\\ a_{1\\ a_{1\\ a_{111\\ a_{11\\ a_{11\\ a_{11\\ a_{11\\ a_{111\\ a_{_{_{_{_{_{_{_{_{_{_{_{_{_{_{_{_{_{_{_{_{_____{_{__{_{_{

Cuando hallamos los determinantes menores, podemos evaluar el determinante de la primera columna, es decir el determinante de la columna donde está, de esta manera tenemos que:

\begin{aligned}\begin{vmatrix}a_{_1}&b_{_1}&c_{_1}\\a_{_2}&b_{_2}&c_{_2}\\a_{_3}&b_{_3}&c_{_3}\end{vmatrix}&=a_{_1}\begin{vmatrix}b_{_2}&c_{_2}\\b_{_3}&c_{_3}\end{vmatrix}-a_{_2}\begin{vmatrix}b_{_1}&c_{_1}\\b_{_3}&c_{_3}\end{vmatrix}+a_{_3}\begin{vmatrix}b_{_1}&c_{_1}\\b_{_2}&c_{_2}\end{vmatrix}\\&\downarrow&\downarrow&\downarrow\\&\boxed{\begin{array}{c}\end{array}}\begin{vmatrix}\begin{array}{c}\end{array}\begin{vmatrix}\end{array}\begin{vmatrix}\end{array}\begin{vmatrix}\end{array}\begin{vmatrix}\end{vmatrix}\begin{array}\end{vmatrix}\begin{vmatrix}\end{vmatrix}\begin{vmatrix}\end{vmatrix}\begin{vmatrix}\end{vmatrix}\begin{vmatrix}\end{vmatrix}\begin{vmatrix}\end{vmatrix}\begin{vmatrix}\end{vmatrix}\begin{vmatrix}\end{vmatrix}\begin{vmatrix}\end{vmatrix}\begin{vmatrix}\end{vmatrix}\begin{vmatrix}\end{vmatrix}\begin{vmatrix}\end{vmatrix}\begin{vmatrix}\end{vmatrix}\begin{vmatrix}\end{vmatrix}\begin{vmatrix}\end{vmatrix}\begin{vmatrix}\end{vmatrix}\begin{vmatrix}\end{vmatrix}\begin{vmatrix}\end{vmatrix}\begin{vmatrix}\end{vmatrix}\begin{vmatrix}\end{vmatrix}\begin{vmatrix}\end{vmatrix}\begin{vmatrix}\end{vmatrix}\begin{vmatrix}\end{vmatrix}\begin{vmatrix}\end{vmatrix}\begin{vmatrix}\end{vmatrix}\begin{vmatrix}\end{vmatrix}\begin{vmatrix}\end{vmatrix}\begin{vmatrix}\end{vmatrix}\begin{vmatrix}\end{vmatrix}\begin{vmatrix}\end{vmatrix}\begin{vmatrix}\end{vmatrix}\begin{vmatrix}\end{vmatrix}\begin{vmatrix}\end{vmatrix}\begin{vmatrix}\end{vmatrix}\begin{vmatrix}\end{vmatrix}\begin{vmatrix}\end{vmatrix}\begin{vmatrix}\end{vmatrix}\begin{vmatrix}\end{vmatrix}\begin{vmatrix}\end{vmatrix}\begin{vmatrix}\end{vmatrix}\begin{vmatrix}\end{vmatrix}\begin{vmatrix}\end{vmatrix}\begin{vmatrix}\end{vmatrix}\begin{vmatrix}\end{vmatrix}\begin{vmatrix}\end{vmatrix}\begin{vmatrix}\end{vmatrix}\begin{vmatrix}\end{vmatrix}\begin{vmatrix}\end{vmatrix}\begin{vmatrix}\end{vmatrix}\begin{vmatrix}\begin{vmatrix}\end{vmatrix}\begin{vmatrix}\end{vmatrix}\begin{vmatrix}\end{vmatrix}\begin{vmatrix}\end{vmatrix}\begin{vmatrix}\end{vmatrix}\begin{vmatrix}\end{vmatrix}\begin{vmatrix}\begin{vmatrix}\end{vmatrix}\begin{vmatrix}\end{vmatrix}\begin{vmatrix}\end{vmatrix}\begin{vmatrix}\end{vmatrix}\begin{vmatrix}\begin{vmatrix}\end{vmatrix}\begin{vmatrix}\end{vmatrix}\begin{vmatrix}\end{vmatrix}\begin{vmatrix}\begin{vmatrix}\end{vmatrix}\begin{vmatrix}\end{vmatrix}\begin{vmatrix}\end{vmatrix}\begin{vmatrix}\begin{vmatrix}\end{vmatrix}\begin{vmatrix}\end{vmatrix}\begin{vmatrix}\begin{vmatrix}\end{vmatrix}\begin{vmatrix}\end{vmatrix}\begin{vmatrix}\end{vmatrix}\begin{vmatrix}\begin{vmatrix}\end{vmatrix}\begin{vmatrix}\begin{vmatrix}\end{vmatrix}\begin{vmatrix}\end{vmatrix}\begin{vmatrix}\begin{vmatrix}\end{vmatrix}\begin{vmatrix}\end{vmatrix}\begin{vmatrix}\begin{vmatrix}\end{vmatrix}\begin{vmatrix}\begin{vmatrix}\end{vmatrix}\begin{vmatrix}\begin{vmatrix}\end{vmatrix}\begin{vmatrix}\end{vmatrix}\begin{vmatrix}\begin{vmatrix}\begin{vmatrix}\end{vmatrix}\begin{vmatrix}\end{vmatrix}\begin{vmatrix}\begin{vmatrix}\end{vmatrix}\begin{vmatrix}\begin{vmatrix}\end{vmatrix}\begin{vmatrix}\begin{vmatrix}\end{vmatrix}\begin{vmatrix}\begin{vmatrix}\end{vmatrix}\begin{vmatrix}\end{vmatrix}\begin{vmatrix}\begin{vmatrix}\end{vmatrix}\begin{vmatrix}\begin{vmatrix}\end{vmatrix}\begin{vmatrix}\begin{vmatrix}\end{vmatrix}\begin{vmatrix}\begin{vmatrix}\end{vmatrix}\begin{vmatrix}\begin{vmatrix}\end{vmatrix}\begin{vmatrix}\begin{vmatrix}\end{vmatrix}\begin{vmatrix}\begin{vmatrix}\end{vmatrix}\begin{vmatrix}\begin{vmatrix}\begin{vmatrix}\end{vmatrix}\begin{vmatrix}\end{vmatrix}\begin{vmatrix}\begin{vmatrix}\begin{vmatrix}\end{vmatrix}\begin{vmatrix}\begin{vmatrix}\end{vmatrix}\begin{vmatrix}\begin{vmatrix}\end{vmatrix}\begin{vmatrix}\begin{vmatrix}\begin{vmatrix}\begin{vmatrix}\end{vmatrix}\begin{vmatrix}\begin{vmatrix}\end{vmatrix}\begin{vmatrix}\end{vmatrix}\begin{vmatrix}\begin{vmatrix}\begin{vmatrix}\begin{vmatrix}\end{vmatrix}\begin{vmatrix}\begin{vmatrix}\end{vmatrix}\begin{vmatrix}\begin{vmatrix}\begin{vmatrix}\end{vmatrix}\begin{vmatrix}\begin{vmatrix}\begin{vmatrix}\end{vmatrix}\begin{vmatrix}\end{vmatrix}\begin{vmatrix}\begin{vmatrix}\begin{vmatrix}\end{vmatrix}\begin{vmatrix}\begin{vmatrix}\begin{vmatrix}\end{vmatrix}\begin{vmatrix}\begin{vmatrix}\end{vmatrix}\begin{vmatrix}\begin{vmatrix}\begin{vmatrix}\begin{vmatrix}\end{vmatrix}\begin{vmatrix}\begin{vmatrix}\end{vmatrix}\begin{vmatrix}\begin{vmatrix}\begin{vmatrix}\begin{vmatrix}\end{vmatrix}\begin{vmatrix}\begin{vmatrix}\begin{vmatrix}\end{vmatrix}\begin{vmatrix}\begin{vmatrix}\begin{vmatrix}\begin{vmatrix}\end{vmatrix}\begin{vmatrix}\begin{vmatrix}\begin{vmatrix}\begin{vmatrix}\begin{vmatrix}\end{vmatrix}\begin{vmatrix}\begin{vmatrix}\begin{vmatrix}\begin{vmatrix}\begin{vmatrix}\begin{vmatrix}\end{vmatrix}\begin{vmatrix}\begin{vmatrix}\end{vmatrix}\begin{vmatrix}\begin{vmatrix}\begin{vmatrix}\begin{vmatrix}\begin{vmatrix}\begin{vmatrix}\end{vmatrix}\begin{vmatrix}\begin{vmatrix}\begin{vmatrix}\begin{vma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Regla de Cramer para sistemas de tres variables.

Mostraremos de forma general como resolver un sistema de ecuaciones de 3\times3 apartir de la regla, después mostraremos un ejemplo que los ilustrara mejor.

Si tenemos un sistema de ecuaciones de 3\times3 como el siguiente:

\begin{array}{l}a_{_1}x+b_{_1}y+c_{_1}z=d_{_1},\\a_{_2}x+b_{_2}y+c_{_2}z=d_{_2},\\a_{_3}x+b_{_3}y+c_{_3}z=d_{_3},\end{array}

Planteamos cada uno de los determinantes del sistema como:

\left|\mathsf{A}\right|=\left|\begin{matrix}a_{1}&b_{1}&c_{1}\\ a_{2}&b_{2}&c_{2}\\ a_{3}&b_{3}&c_{3}\end{matrix}\right|\left|\mathsf{A}_{x}\right|=\left|\begin{matrix}d_{1}&b_{1}&c_{1}\\ d_{2}&b_{2}&c_{2}\\ d_{3}&b_{3}&c_{3}\end{matrix}\right|\left|\mathsf{A}_{y}\right|=\left|\begin{matrix}a_{1}&d_{1}&c_{1}\\ a_{2}&d_{2}&c_{2}\\ a_{3}&d_{3}&c_{3}\end{matrix}\right|\left|\mathsf{A}_{z}\right|=\left|\begin{matrix}a_{1}&b_{1}&d_{1}\\ a_{2}&b_{2}&d_{2}\\ a_{3}&b_{3}&d_{3}\end{matrix}\right|

Matemáticas· Grado 9


46