Advanced Microeconomics/Linear Algebra

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Cramer's Rule[edit]

For a three by three system:

\begin{array}{c}
ax+by+cz=j\\
dx+ey+fz=k\\
gx+hy+iz=l\\
\end{array}
In matrix form:

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

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}} 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}}

In general: Ax = b
 x_i = \frac{|A_i|}{|A|} \; \forall i= \{1,\dots,n\} Where A_i is formed by replacing the vector associated with x_i by the column vector b.