System of Three Linear Equations in Matrix Form

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

A System of Three Linear Equations
Consider the system of three equations of straight lines is 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}

Equations (i), (ii) and (iii) are linear equations and three variables, x and y. can be written in matrix form as follows:

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}

Here 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 the 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.