Cramer’s Rule for invertible $2\times 2$ matrices

• Consider the following system of linear equations of two variables:

  $a_{11}x_1+a_{12}x_2=b_1$

  $a_{21}x_1+a_{22}x_2=b_2$

• Matrix Representation : $AX=b$ ; $A=\begin{bmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{bmatrix}$, $b=\begin{bmatrix} b_1\\b_2\end{bmatrix}$ . (Assuming $A$ is invertible).

• Define $A_{x_1}= \begin{bmatrix} b_1 & a_{12} \\ b_2 & a_{22}\end{bmatrix}$$, A_{x_2} = \begin{bmatrix} a_{11} & b_1 \\ a_{21} & b_2 \end{bmatrix}$

• Then the solution of the system of equations in 2 variables is:

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

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

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Cramer’s Rule for invertible $3\times 3$ matrices

• Consider the following system of linear equations in 3 variables:

  $a_{11}x_1+a_{12}x_2+a_{13}x_3=b_1$

  $a_{21}x_1+a_{22}x_2+a_{23}x_3=b_2$

  $a_{31}x_1+a_{32}x_2+a_{33}x_3=b_3$

• The solution of the system of equations of 3 variables:

  $\large x_1=\frac {det(A_{x_1})} {det(A)}$ ; $\large x_2=\frac {det(A_{x_2})} {det(A)}$ ; $\large x_3=\frac {det(A_{x_3})} {det(A)}$

Cramer’s Rule for invertible $n\times n$ matrices

The unique solution of system of linear equations with n variables:

$x_i=\frac {det(A_{x_i})} {det(A)}$