# Show that the given equation have infinitely many solutions.

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

This is because $$\det A=\triangle=0$$ only ensures that $$\mathrm{rk}(A), and $$\triangle_i=0$$ for each $$i$$ only ensures that the complete matrix $$A'$$ has $$\mathrm{rk}(A') 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.
• One last question, what is $A'$ here? Aug 23, 2020 at 10:20
• $A'$ is the so-called Augmented matrix Aug 23, 2020 at 10:22