List of mathematical formulas: “Cramer's rule. Matrix method for solving the system

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.

  1. 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}
    ]
  2. 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}
    ]
  3. Vector of free terms:
    [
    \mathbf{d} = \begin{pmatrix}
    d_1 \
    d_2 \
    \vdots \
    d_n
    \end{pmatrix}
    ]
  4. Determinant of the coefficient matrix:
    [
    D = \det(A)
    ]
  5. 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.
  1. 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.

От Math

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