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Cramer's Rule

Cramer's rule is a formula for solving systems of linear equations using determinants. It expresses the solution for a particular variable in terms of the determinants of matrices derived from the coefficients of the equations and their corresponding constants.

Forumla

Cramer's Rule for 2x2 Systems

  • \(\begin{align} A &= \begin{bmatrix} a & b \\ c & d \end{bmatrix} & B &= \begin{bmatrix} e \\ f \end{bmatrix} \end{align}\)

  • \(\begin{align} A_{x1} &= \begin{bmatrix} e & b \\ f & d \end{bmatrix} & A_{x2} &= \begin{bmatrix} a & e \\ c & f \end{bmatrix} \end{align}\)

  • \(\begin{align} x_1 &= \frac{\det(A_{x1})}{\det(A)} & x_2 &= \frac{\det(A_{x2})}{\det(A)} \end{align}\)

This is how we find the solution to a system of linear equations using Cramer's rule.

Cramer's Rule for 3x3 Systems

  • \(\begin{align} A &= \begin{bmatrix} a & b & c \\ d & e & f \\ g & h & i \end{bmatrix} & B &= \begin{bmatrix} j \\ k \\ l \end{bmatrix} \end{align}\)
  • \(\begin{align} A_{x1} &= \begin{bmatrix} j & b & c \\ k & e & f \\ l & h & i \end{bmatrix} & A_{x2} &= \begin{bmatrix} a & j & c \\ d & k & f \\ g & l & i \end{bmatrix} & A_{x3} &= \begin{bmatrix} a & b & j \\ d & e & k \\ g & h & l \end{bmatrix} \end{align}\)
  • \(\begin{align} x_1 &= \frac{\det(A_{x1})}{\det(A)} & x_2 &= \frac{\det(A_{x2})}{\det(A)} & x_3 &= \frac{\det(A_{x3})}{\det(A)} \end{align}\)

Note

  • A 3x3 matrix is invertible if the determinant is not equal to 0.
  • If the determinant is not equal to 0, then the inverse of the matrix is equal to the adjoint divided by the determinant.