Cramer’s rule

Cramer’s rule is used when solving the system

\[Ax = b\]

As seen in Determinant, determinants can be viewed as area(or volume) of a matrix.

\[x = \begin{bmatrix} x_1 \\ x_2 \end{bmatrix}, \: b = \begin{bmatrix} b_1 \\ b_2 \end{bmatrix}\] \[det( \begin{bmatrix} 1 & x_1 \\ 0 & x_2 \end{bmatrix} ) = Area(\begin{bmatrix} 1 & x_1 \\ 0 & x_2 \end{bmatrix}) = x_2\]

As linear transformation \(A\) multiplicates the area of a matrix by \(det(A)\), the above equation can be adjusted as

\[det(A \begin{bmatrix} 1 & x_1 \\ 0 & x_2 \end{bmatrix} ) = det( \left[ \begin{array}{c|c} A \begin{bmatrix} 1 \\ 0 \end{bmatrix} & A \begin{bmatrix} x_1 \\ x_2 \end{bmatrix} \end{array}\right] ) = det(A)x_2\]

which yields \(x_2 = \dfrac{det(\left[ \begin{array}{c|c} A \begin{bmatrix} 1 \\ 0 \end{bmatrix} & \begin{bmatrix} b_1 \\ b_2 \end{bmatrix} \end{array}\right] )}{det(A)}\)

Note that \(det(\left[ \begin{array}{c|c} A \begin{bmatrix} 1 \\ 0 \end{bmatrix} & \begin{bmatrix} b_1 \\ b_2 \end{bmatrix} \end{array}\right] )\) is matrix \(A\), with it’s \(2nd\) column replaced by \(b\).

Similarly, \(x_1 = \dfrac{det(\left[ \begin{array}{c|c} \begin{bmatrix} b_1 \\ b_2 \end{bmatrix} & A \begin{bmatrix} 0 \\ 1 \end{bmatrix} \end{array}\right] )}{det(A)}\)

The generalized cramer’s rule in \(n\)-dimensional space is

\[x_j = \dfrac{det(B_j)}{det(A)} \\ where \: B_j = \begin{bmatrix} a_{11} & \cdots & b_1 & \cdots & a_{1n} \\ a_{21} & \cdots & b_2 & \cdots & a_{2n} \\ \vdots & \ddots & \vdots & \ddots & \vdots \\ a_{n1} & \cdots & b_n & \cdots & a_{nn} \end{bmatrix}\]