7.2 Equivalent Statements

Proving Non-Conditional Statements

Theorem

• Suppose $A$ is an $n\times n$ matrix. The following statements are equivalent:
  • (a) The matrix $A$ is invertible.
  • (b) The equation $Ax=b$ has a unique solution for every $b\in Rn$.
  • (c) The equation $Ax=0$ has only the trivial solution.
  • (d) The reduced row echelon form of $A$ is In.
  • (e) $det(A)\ne 0$.
  • (f) The matrix $A$ does not have 0 as an eigenvalue.
• One approach to proving the theorem about the n × n matrix would be to prove the conditional statement ($a)\Rightarrow (b$), then $(b)\Rightarrow (c)$, then $(c)\Rightarrow (d)$, then $(d)\Rightarrow (e)$, then ($e)\Rightarrow (f)$ and finally $(f)\Rightarrow (a)$.