Solution Set and Rank

From a system of $m$ linear equations having $n$ unknowns, we can create a coefficient matrix $A_{m \times n}$ and an augmented matrix $[A|b]_{m \times n+1}$. The rank of any matrix is equal to the number of linearly independent rows of that matrix, which is equal to the number of pivots in the $rref$ (reduced row echelon form) of that matrix. We saw earlier that there are three possibilities for the solution set of a system of linear equations, including no solution, a unique solution, or infinite solutions. This lesson explores these three possibilities’ relationships with rank.

Nonhomogeneous systems

We first consider a nonhomogeneous system, $A\bold{x}=\bold{b}$ where at least one element of vector $\bold b$ is nonzero.

No solution

A system of linear equations has no solution when it’s inconsistent. We can employ rank to find this inconsistency. A system $A\bold{x}=\bold{b}$ has no solution if and only if $r(A)<r([A|b])$. That is, the rank of the coefficient matrix is less than the rank of the augmented matrix. In the following example, the system has no solutions, because the rank of the coefficient matrix is $1$, whereas the rank of the augmented matrix is $2$. The rank is computed by first converting each matrix to its $rref$ and subsequently counting the number of nonzero rows.

Linear system	Coefficient matrix’s $rref$	Augmented matrix’s $rref$
$\begin{array}{} x_1+2x_2=3 \\ 2x_1+4x_2=7 \end{array}$	$\begin{pmatrix}1&2\\ 0&0\end{pmatrix}$	$\begin{pmatrix}1&2&3\\ 0&0&1\end{pmatrix}$

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