System of Three Linear Equations in Matrix Form

In this tutorial we shall discuss system of three linear equations in matrix form

A System of Three Linear Equations:
Consider the system of three equations of straight lines are given as

\begin{gathered} {a_1}x + {b_1}y + {c_1} = 0\,\,\,\,{\text{  -  -   - }}\left( {\text{i}} \right) \\ {a_2}x + {b_2}y + {c_2} = 0\,\,\,\,{\text{  -  -   - }}\left( {{\text{ii}}} \right) \\ {a_3}x + {b_3}y + {c_3} = 0\,\,\,\,{\text{  -  -   - }}\left( {{\text{iii}}} \right) \\ \end{gathered}


Equation (i), (ii) and (iii) are linear equations in three variables x and y can be written as in matrix form as follows:
Now equation (i) becomes

  \Rightarrow \left[ {\begin{array}{*{20}{c}} {{a_1}x + {b_1}y + {c_1}} \\ {{a_2}x + {b_2}y + {c_2}} \\ {{a_3}x + {b_3}y + {c_3}} \end{array}}  \right] = \left[ {\begin{array}{*{20}{c}} 0 \\ \begin{gathered} 0 \\ 0 \\ \end{gathered} \end{array}}  \right]


It can be further written as

\begin{gathered} \Rightarrow \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]\left[ {\begin{array}{*{20}{c}} x \\ y \\ z \end{array}}  \right] = \left[ {\begin{array}{*{20}{c}} 0 \\ 0 \\ 0 \end{array}}  \right] \\ AX = C \\ \end{gathered}


Where A = \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] is the coefficient matrix and X = \left[  {\begin{array}{*{20}{c}} x \\ y \\ z \end{array}} \right] is variable matrix.
We have already discussed that the three lines (i), (ii) and (iii) will be concurrent if

\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| = 0


This shows that the given lines (i), (ii) and (iii) will be concurrent if the coefficient matrix A = \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] is singular.

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