Formulas

Matrices and Determinants

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Systems of Linear Equations

  • Matrices and vectors: \(A\), \(B\), \(X\)
    Coefficients of equations: \({a_{ij}},\) \({a_i},\) \({b_i},\) \({c_i},\) \({d_i}\)
    Inverse matrix: \(A^{-1}\)
    Determinants: \(D,\) \({D_x},\) \({D_y},\) \({D_z}\)
    Unknown variables: \(x,\) \(y,\) \(z,\) \({x_1},\) \({x_2},\) \(\ldots\)
    Natural numbers: \(n,\) \(i,\) \(j\)
    1. Solution of a system of equations in two variables by the Cramer’s rule
      Given a system of two linear equations with two unknowns:
      \( \left\{ \begin{aligned} {a_1}x + {b_1}y &= {d_1} \\ {a_2}x + {b_2}y &= {d_2} \end{aligned} \right. \).

      The solution of this system is expressed by the formulas
      \(x = {\large\frac{{{D_x}}}{D}\normalsize},\;\) \(y = {\large\frac{{{D_y}}}{D}\normalsize}\;\) (Cramer’s formulas),
      where the determinants \(D\), \({D_x}\), \({D_y}\) are given by
      \(D = \left| {\begin{array}{*{20}{c}} {{a_1}}&{{b_1}}\\ {{a_2}}&{{b_2}} \end{array}} \right| =\) \( {a_1}{b_2} – {a_2}{b_1},\;\) \({D_x} = \left| {\begin{array}{*{20}{c}} {{d_1}}&{{b_1}}\\ {{d_2}}&{{b_2}} \end{array}} \right| =\) \({d_1}{b_2} – {d_2}{b_1},\;\) \({D_y} = \left| {\begin{array}{*{20}{c}} {{a_1}}&{{d_1}}\\ {{a_2}}&{{d_2}} \end{array}} \right| =\) \( {a_1}{d_2} – {a_2}{d_1}.\)
    2. Special cases of the solutions of a system of equations in two variables
      – If \(D \ne 0\), then the system is consistent and has the single solution \(x = {\large\frac{{{D_x}}}{D}\normalsize},\;\) \(y = {\large\frac{{{D_y}}}{D}\normalsize};\)
      – If \(D = 0\) and \({D_x} \ne 0\) (or \({D_y} \ne 0\)), then the system is inconsistent (has no solutions);
      – If \(D = {D_x} \) \(= {D_y} \) \(= 0\), then the system is consistent and has an infinite set of solutions.
    3. Solution of a system of equations in three variables by the Cramer’s rule
      Consider a system of \(3\) equations with \(2\) unknowns:
      \( \left\{ \begin{aligned} {a_1}x + {b_1}y + {c_1}z &= {d_1} \\ {a_2}x + {b_2}y + {c_2}z &= {d_2} \\ {a_3}x + {b_3}y + {c_3}z &= {d_3} \end{aligned} \right. \).

      The solution of this system is determined by the Cramer’s formulas:
      \(x = {\large\frac{{{D_x}}}{D}\normalsize},\;\) \(y = {\large\frac{{{D_y}}}{D}\normalsize},\;\) \(z = {\large\frac{{{D_z}}}{D}\normalsize},\)
      where the determinants \(D\), \({D_x}\), \({D_y}\), \({D_z}\) are given by
      \(D = \left| {\begin{array}{*{20}{c}} {{a_1}}&{{b_1}}&{{c_1}}\\ {{a_2}}&{{b_2}}&{{c_2}}\\ {{a_3}}&{{b_3}}&{{c_3}} \end{array}} \right|,\;\) \({D_x} = \left| {\begin{array}{*{20}{c}} {{d_1}}&{{b_1}}&{{c_1}}\\ {{d_2}}&{{b_2}}&{{c_2}}\\ {{d_3}}&{{b_3}}&{{c_3}} \end{array}} \right|,\;\) \({D_y} = \left| {\begin{array}{*{20}{c}} {{a_1}}&{{d_1}}&{{c_1}}\\ {{a_2}}&{{d_2}}&{{c_2}}\\ {{a_3}}&{{d_3}}&{{c_3}} \end{array}} \right|,\;\) \({D_z} = \left| {\begin{array}{*{20}{c}} {{a_1}}&{{b_1}}&{{d_1}}\\ {{a_2}}&{{b_2}}&{{d_2}}\\ {{a_3}}&{{b_3}}&{{d_3}} \end{array}} \right|.\)
    4. Special cases of the solutions of a system of equations in three variables
      – If \(D \ne 0\), then the system is consistent and has the single solution \(x = {\large\frac{{{D_x}}}{D}\normalsize},\;\) \(y = {\large\frac{{{D_y}}}{D}\normalsize}\;\) \(z = {\large\frac{{{D_z}}}{D}\normalsize} ;\)
      – If \(D = 0\) and \({D_x} \ne 0\) (or \({D_y} \ne 0\) or \({D_z} \ne 0\)), then the system is inconsistent (i.e. it has no solutions);
    5. Linear system of \(n\) equations with \(n\) unknowns in matrix form
      A system of linear equations of order \(n\)
      \( \left\{ \begin{aligned} {a_{11}}{x_1} + {a_{12}}{x_2} + \ldots + {a_{1n}}{x_n} &= {b_1} \\ {a_{21}}{x_1} + {a_{22}}{x_2} + \ldots + {a_{2n}}{x_n} &= {b_2} \\ \ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots \\ {a_{n1}}{x_1} + {a_{n2}}{x_2} + \ldots + {a_{nn}}{x_n} &= {b_n} \end{aligned} \right. \)
      can be written in matrix form:
      \(\left[ {\begin{array}{*{20}{c}} {{a_{11}}}&{{a_{12}}}& \ldots &{{a_{1n}}}\\ {{a_{21}}}&{{a_{22}}}& \ldots &{{a_{2n}}}\\ \vdots & \vdots &{}& \vdots \\ {{a_{n1}}}&{{a_{n2}}}& \ldots &{{a_{nn}}} \end{array}} \right] \cdot \) \(\left[ {\begin{array}{*{20}{c}} {{x_1}}\\ {{x_2}}\\ \vdots \\ {{x_n}} \end{array}} \right] =\) \( \left[ {\begin{array}{*{20}{c}} {{b_1}}\\ {{b_2}}\\ \vdots \\ {{b_n}} \end{array}} \right],\)
      or in more compact form:
      \(AX = B\),
      where we use the notation
      \(A =\) \( \left[ {\begin{array}{*{20}{c}} {{a_{11}}}&{{a_{12}}}& \ldots &{{a_{1n}}}\\ {{a_{21}}}&{{a_{22}}}& \ldots &{{a_{2n}}}\\ \vdots & \vdots &{}& \vdots \\ {{a_{n1}}}&{{a_{n2}}}& \ldots &{{a_{nn}}} \end{array}} \right],\;\) \(X = \left[ {\begin{array}{*{20}{c}} {{x_1}}\\ {{x_2}}\\ \vdots \\ {{x_n}} \end{array}} \right],\;\) \(B = \left[ {\begin{array}{*{20}{c}} {{b_1}}\\ {{b_2}}\\ \vdots \\ {{b_n}} \end{array}} \right].\)
    6. The solution of a system of \(n\) equations with \(n\) unknowns has the form
      \(X = A^{-1}B\),
      where \(A^{-1}\) is the inverse matrix.