When Does the Equation Ax=B Have Unique, Multiple, or No Solutions?

[MathematicalPhysicist]

i need to remember when does Ax=B has a unique solution, more than one, and no solution?

i think that it has a unique solution when A is non singular i.e when det(A) isn't equal zero (at least that's what's written in my notes).

what about more than one solution, what conditipns should be met for either of A,x or b (also for no solution).

i guess that when A isn't non singular then the equation doesn't have solution but I am not sure, forgot about this.
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[doodle]

There is a unique solution if A is full rank, infinite solutions if A is rank-deficient and b lies in the column space (or range) of A, and no solution if A is rank-deficient and b lies in the left-nullspace of A.

The determinant of a matrix is the product of its eigenvalues. A rank-deficient matrix would have some of its eigenvalues equal zero, thus the determinant equals zero. Any matrix A with det(A) = 0 would therefore admit either infinite number of solutions or no solution for Ax = b.