Show that the given equation have infinitely many solutions.

Question

Consider $m$ linear equations of $m$ variables. Let's show it using matrices. $A=(a_{ij})_{1\le i,j \le m}$ $X=\begin{pmatrix} x_1 \\ x_2 \\x_3 \\..\\x_m \end{pmatrix} $

$B=\begin{pmatrix} b_1 \\ b_2 \\.. \\b_m \end{pmatrix} $ It is given that, $det(A)=0$. Which condition will ensure that the given set of equations have infinitely many solutions? In Cramer's rule ,I read from Wikipedia that $\Delta =\Delta_{1}=\Delta_{2}=...=\Delta_{m}=0$ will not ensure that the given set of equations have infinitely many solution. Thank you in advance. Your help will be highly appreciated. The Wikipedia page is the following https://en.m.wikipedia.org/wiki/Cramer%27s_rule

Answer 1

This is because $\det A=\triangle=0$ only ensures that $\mathrm{rk}(A)<m$, and $\triangle_i=0$ for each $i$ only ensures that the complete matrix $A'$ has $\mathrm{rk}(A')<m,$ but we don't yet know if $\mathrm{rk}(A)=\mathrm{rk}(A'),$ that is necessary to have a compatible linear system.

A linear system has infinitely many solutions if it is compatible ($\mathrm{rk}(A)=\mathrm{rk}(A')$) and the number of unknowns is stricly greater than the common rank.