Does being a n x n homogeneous equation with a nontrivial solution mean A must have fewer than n pivots?

Question

Why is it the case that being a homogeneous equation (Ax = 0) where A is an n x n matrix guarantees that A must have fewer than n pivots? Certainly there exists a matrix that would not allow this?

Please explain this in plain English

Answer 1

The number of pivots is the rank of the matrix.

Therefore, for a $n \times n$ matrix, having $n$ pivots is equivalent to have full rank, i.e. to be invertible, i.e. to have null space equal to $\lbrace 0 \rbrace$, i.e. to have only the trivial solution $X=0$ to the system $AX=0$.

In other words, if the rank is strictly less than $n$, then the rank of the matrix is strictly less than $n$, so the null space has dimension at least $1$, so there exists at least one non-trivial vector $X$ such that $AX=0$.