# 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

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

It can be further written as

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

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.