Consistency of a System of Linear Equations

Setting Up the System in Matrix Form

A system of linear equations such as

can be written compactly as a single matrix equation $AX=B$, where

$A$ is called the coefficient matrix. A system is called consistent if it has at least one solution (whether exactly one, or infinitely many), and inconsistent if it has no solution at all.

The Key Test: $|A|$

Case 1: $|A|\ne0$. The coefficient matrix is non-singular, so $A^{-1}$ exists (Section 6), and $X=A^{-1}B$ is the unique solution of the system (Section 8 works through this method in full). A system with $|A|\ne0$ is therefore always consistent, with exactly one solution -- this is the case the syllabus focuses on for the matrix-inverse solving method.

Case 2: $|A|=0$. Here $A^{-1}$ does not exist, so the inverse method cannot be used directly, and the system may be either inconsistent or consistent-with-infinitely-many-solutions. Distinguishing between these two possibilities requires examining the matrix product $(\operatorname{adj}A)B$:

• If $(\operatorname{adj}A)B \ne O$ (the zero matrix) -- equivalently, if at least one of the determinants formed by replacing a column of $A$ with $B$ is non-zero -- the system is inconsistent: it has no solution.
• If $(\operatorname{adj}A)B = O$, the system is consistent, but has infinitely many solutions rather than a unique one.

A full treatment of the second sub-case (deciding which infinitely-many-solutions situation applies) needs tools beyond this chapter's scope; here it is enough to recognise that $|A|=0$ signals "not a unique solution" and to identify, via $(\operatorname{adj}A)B$, whether that means no solution or infinitely many.

A Quick Two-Variable Illustration

For two equations in two unknowns, $a_1x+b_1y=c_1,\ a_2x+b_2y=c_2$, define the three determinants

$D=\begin{vmatrix}a_1&b_1\\a_2&b_2\end{vmatrix},\quad D_1=\begin{vmatrix}c_1&b_1\\c_2&b_2\end{vmatrix},\quad D_2=\begin{vmatrix}a_1&c_1\\a_2&c_2\end{vmatrix}.$ …