Why are the columns of a matrix linearly dependent if and only if the determinant is 0?

The columns of a matrix are linearly dependent if at least one column can be expressed as a linear combination of the others. When the determinant of a square matrix is zero, it indicates that the matrix is not invertible, which implies that its columns do not span the entire vector space. Consequently, the existence of a non-trivial solution to the equation Ax = 0 shows that the columns are linearly dependent. Thus, a determinant of zero is both a necessary and sufficient condition for linear dependence of the matrix's columns.