System of linear equations-unique, infinitely many, or no solutions

[kingwinner]

Suppose we have a system of 2 equations in 2 unknowns x,y:
a₁x+b₁y=c₁
a₂x+b₂y=c₂

If the determinant of
[a1 b1
a2 b2]
is nonzero, then the solution to the system exists and is unique. [I am OK with this]

If the determinant of
[a1 b1
a2 b2]
is zero,
this does not distinguish between the cases of no solution and infinitely many solutions.

To gain some insight, we need to further check the determinatnts of
[a1 c1
a2 c2]
and
[b1 c1
b2 c2]
If both determinants are zero, then the system has infinitely many soltuions.
If at least one of the two determinants are nonzero, then the system has no solution.
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Is the bolded part correct or not? I don't see why it is true. How can we prove it?


Thanks for any help!

 

[kingwinner]

Can someone "confirm or disprove" the bolded part, please?

[my PDE book seems to be applying these ideas from linear algebra to study the solvability of initial value problems for quasilinear partial differential equations, but I can't find those results in my linear algebra textbooks other than the result "det(A) is not 0 <=> solution exists and is unique"]