In this section, we shall discuss application of determinants and matrices for solving the system of linear equations in two or three variables and for checking the consistency of the system of linear equations.

Consistent system: A system of equations is said to be consistent if its solution (one or more) exists. Inconsistent system: A system of equations is said to be inconsistent if its solution does not exist.

Solution of system of linear equations using inverse of a matrix

Let us express the system of linear equations as a matrix equation and solve it using inverse of the coefficient matrix. Consider the system of equations \(a_1x + b_1y + c_1z = d_1\) \(a_2x + b_2y + c_2z = d_2\) \(a_3x + b_3y + c_3z = d_3\) Let \(A = \begin{bmatrix} a_1 & b_1 & c_1 \\ a_2 & b_2 & c_2 \\ a_3 & b_3 & c_3 \end{bmatrix}\), \(X = \begin{bmatrix} x \\ y \\ z \end{bmatrix}\) and \(B = \begin{bmatrix} d_1 \\ d_2 \\ d_3 \end{bmatrix}\). Then, the given system of equations can be written as, \(AX = B\).