Cramer's rule

Nevertheless, Cramer's rule is useful in a certain way, because the solution of a system of equations depends continuously on the coefficients of the system of equations. In this way, one can obtain estimates for the individual components of the solution.

Procedure

Linear system of equations

$\def\arraystretch{1.25} \begin{array}{ccccc}ax_1+bx_2+cx_3= b_1\\dx_1+ex_2+fx_3= b_2\\gx_1+hx_2+ix_3= b_3\end{array}$

The linear system of equations is transformed into an extended coefficient matrix.

$\def\arraystretch{1.25} \;\;\Rightarrow\;\;( A\left| b)\right.=\left(\begin{array}{ccc} a& b& c\\ d& e& f\\ g& h& i\end{array} \left| \begin{array}{c} b_1\\ b_2\\ b_3\end{array}\right.\right)$

Calculation of $x_1,\;x_2,\;x_3$ :

$\def\arraystretch{1.25} \begin{array}{l}A=\begin{pmatrix}a& b & c\\ d & e & f\\ g & h & i \end{pmatrix}\Rightarrow\text{det}\;A=\text{det }\begin{pmatrix}a& b& c\\ d & e & f\\g & h & i \end{pmatrix}\end{array}$

The determinant is calculated.

$\text{det}(A_{x_1}) =\text{det }\begin{pmatrix}b_1 & b & c\\ b _2 & e& f\\ b _3& h & i\end{pmatrix}$

The column of $x _1$ values is replaced by the result column and the determinant is calculated from it.

$\text{det}(A_{x_2}) =\text{det }\begin{pmatrix} a & b_1 & c\\ d & b _2 & f\\ g & b _3 & i\end{pmatrix}$

The column of $x_2$ values is replaced by the result column and the determinant is calculated from it.

$\text{det}(A_{x_3}) =\text{det }\begin{pmatrix} a & b & b_1 \\ d & e &b _2 \\ g & h &b _3\end{pmatrix}$

The column of $x_3$ values is replaced by the result column and the determinant is calculated from it.

If $\det\left(A\right)\ \ne0$, the solutions $x_1,\;x_2,\;x_3$ of the given linear system of equations are then obtained as follows:

$x_1=\frac{\text{det}(A_{x_1})}{\det(A)}$

$x_2=\frac{\text{det}(A_{x_2})}{\det(A)}$

$x_3=\frac{\text{det}(A_{x_3})}{\det(A)}$