Hello, I'd like to know if the following two paragraphs regarding the determinant of a matrix are correct and also, am I missing any other important implications by calculating the determinant? any other important things I can find from with that value? thanks. 1. If det A=0 <=> Linear Dependence <=> Infinitely many solutions (hence non trivial solution) <=> non invertible (or singular) matrix <=> vectors are parallel. 2. If det A != 0 <=> L.I <=> unique solution <=> invertible (also nonsingular or regular) matrix
If the determinant is zero and at least one of the determinants in the numerator (using Cramer's rule) is not 0, then the system is inconsistent and there is no solution. However, if the determinant is 0 and all the determinants in the numerators are 0, then the system is dependent with infinitely many solutions.