Final answer:
The question deals with linear algebra and the unique solvability of the equation Ax = 0 when A is an invertible matrix. The only solution in this case is the null vector x = 0, affirming the matrix's full rank and the linear independence of its columns.
Step-by-step explanation:
The subject in question pertains to linear algebra, specifically the properties of matrices and their invertibility. When all the pivot variables in a matrix are non-zero, it suggests that each column of the matrix is a pivot column, indicating that the matrix is invertible. An invertible matrix has a unique solution to the equation Ax = 0, which is the null vector x = 0. This is because an invertible matrix has full rank, meaning its columns are linearly independent, and thus the only vector that can be represented as a linear combination of these columns resulting in the zero vector is the null vector itself.
Understanding the role of the null vector is significant. It is the vector equivalent of the number zero; it has components of zero and demonstrates no direction or magnitude. In vector equations, when the difference between two vectors equals the null vector, it indicates that those two vectors are equal.