Invertible Matrix Theorem

A list of conditions on a square matrix that all hold together, or all fail together.

For an $n\times n$ matrix $A$ , the following are logically equivalent: (1) $A$ is invertible; (2) $A$ is row equivalent to $I_n$ ; (3) $A\mathbf x=\mathbf 0$ has only the trivial solution; (4) $A\mathbf x=\mathbf b$ has exactly one solution for every $\mathbf b$ . Later chapters add more equivalent conditions (determinant, rank, column space) to the same list — for a given $A$ , either all hold or all fail, never a mix.