If the determinant $\det(A)$ of the matrix $A$ of a non-homogeneous system of equations is $0$, then how do we know if it has no solutions or infinitely many solutions?
And while we are at it, kindly answer the following "sub-questions" arising from it. I shall be really grateful to you as it will be crucial to my understanding of the whole thing:
a) Since the determinant being zero means that a situation of "Division by zero" arises (using Cramer's Rule), the "no solution" option is understandable as division by zero is not defined. But it confuses me how then, in any circumstance, the system can have infinitely many solutions. I mean, won't we encounter division by zero in all cases when determinant is zero? So please give me an intuitive and insightful explanation to it.
b) Will I be wrong to assume that, in a case when determinant is equal to zero, there are infinitely many solutions if and only if it's a homogeneous system of equations? Please explain why or why not.
And kindly don't forget the main question--"for determinant $=0$, how to know if there are no or infinitely many solutions?"