List of formulas related to Cramer’s rule and the matrix method for solving systems of linear equations.
Kramer’s rule
Cramer’s rule is used to solve a system of linear equations when the number of equations is equal to the number of unknowns and the determinant of the coefficient matrix is not equal to zero.
- System of linear equations:
[
\begin{cases}
a_1x_1 + b_1x_2 + \ldots + c_1x_n = d_1 \
a_2x_1 + b_2x_2 + \ldots + c_2x_n = d_2 \
\vdots \
a_nx_1 + b_nx_2 + \ldots + c_nx_n = d_n
\end{cases}
] - Matrix of coefficients:
[
A = \begin{pmatrix}
a_1 & b_1 & \ldots & c_1 \
a_2 & b_2 & \ldots & c_2 \
\vdots & \vdots & \ddots & \vdots \
a_n & b_n & \ldots & c_n
\end{pmatrix}
] - Vector of free terms:
[
\mathbf{d} = \begin{pmatrix}
d_1 \
d_2 \
\vdots \
d_n
\end{pmatrix}
] - Determinant of the coefficient matrix:
[
D = \det(A)
] - Determinants for finding variables:
- Для (x_1):
[
D_1 = \det\begin{pmatrix}
d_1 & b_1 & \ldots & c_1 \
d_2 & b_2 & \ldots & c_2 \
\vdots & \vdots & \ddots & \vdots \
d_n & b_n & \ldots & c_n
\end{pmatrix}
] - For (x_2):
[
D_2 = \det\begin{pmatrix}
a_1 & d_1 & \ldots & c_1 \
a_2 & d_2 & \ldots & c_2 \
\vdots & \vdots & \ddots & \vdots \
a_n & d_n & \ldots & c_n
\end{pmatrix}
] - And so on for the remaining variables.
- Formulas for variables:
[
x_i = \frac{D_i}{D}, \quad i = 1, 2, \ldots, n
]
Note
Cramer’s rule works only if the determinant of the matrix (D) is not zero. If (D = 0), the system either has no solutions or has infinitely many solutions.