# 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

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

It can be further written as

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

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.